Linear Equations and Inequalities

Equality and the Balance Model

In maths, an equation is a number sentence that uses the equals sign, \(=\). The equals sign means that the value on the left side is exactly the same as the value on the right side.

For example, in \(7 + 3 = 10\), both sides have the same value. The left side equals 10, and the right side also equals 10. That is why the equation is true.

A helpful way to understand equality is the balance model. Imagine a balance scale. If both sides weigh the same, the scale stays level. If one side changes and the other side does not, the scale tips. Equations work the same way.

When we solve equations, our goal is to keep the equation balanced while finding the value of the unknown number, called the variable. A variable is usually shown with a letter such as \(x\), \(n\), or \(y\).

Key idea: If you do the same operation to both sides of an equation, the equation stays equal.

This idea is called the Properties of Equality. In 7th Grade, the most important ones are:

• Addition Property of Equality: You can add the same number to both sides.
• Subtraction Property of Equality: You can subtract the same number from both sides.
• Multiplication Property of Equality: You can multiply both sides by the same number.
• Division Property of Equality: You can divide both sides by the same nonzero number.

These properties help us keep the balance while we isolate the variable. To isolate the variable means to get the variable by itself on one side of the equation.

Let us connect this to the balance model.

If a scale shows:

you can think of the left side as a box with unknown weight \(x\) and 4 extra units, and the right side as 9 units. To keep the scale balanced, if you remove 4 from the left side, you must also remove 4 from the right side.

The equation stays balanced, and now we know that \(x = 5\).

Why inverse operations matter

An inverse operation undoes another operation.

• Addition and subtraction are inverses.
• Multiplication and division are inverses.

When solving an equation, we use inverse operations to undo what is happening to the variable.

For example:

• If the equation is \(x + 6 = 14\), subtract 6 from both sides.
• If the equation is \(x - 3 = 8\), add 3 to both sides.
• If the equation is \(5x = 20\), divide both sides by 5.
• If the equation is \(\frac{x}{4} = 7\), multiply both sides by 4.

Important rule: Never do something to only one side of the equation. That would break the balance.

For example, if you start with \(x + 2 = 10\) and subtract 2 from only the left side, you would get \(x = 10\), which is not true. The correct step is:

Now the equation is still balanced.

Worked Example 1: One-step equation using subtraction

Solve:

The variable has 7 added to it. To undo that, subtract 7 from both sides.

Check: Substitute 8 for \(x\).

This is true, so the solution is correct.

Worked Example 2: One-step equation using addition

Solve:

The variable has 5 subtracted from it. Use the inverse operation, so add 5 to both sides.

Check:

This is true.

Worked Example 3: One-step equation using division

Solve:

The variable is multiplied by 6. Undo that by dividing both sides by 6.

Check:

The solution is correct.

Worked Example 4: A two-step equation

Solve:

We want to isolate \(x\). Follow the order of operations backward.

First, undo the +3 by subtracting 3 from both sides.

Next, undo the multiplication by 2 by dividing both sides by 2.

Check:

This is true, so \(x = 4\).

How to solve equations with the balance model

1. Look at the variable and decide what is being done to it.
2. Use the inverse operation to undo it.
3. Do the same operation to both sides.
4. Keep going until the variable is alone.
5. Check your answer by substituting it back into the original equation.

Common mistakes to avoid

• Only changing one side: Always do the same thing to both sides.
• Using the wrong inverse operation: If you see addition, use subtraction. If you see multiplication, use division.
• Forgetting to check: Substituting your answer back into the equation helps catch mistakes.

Equality and true equations

An equation is true when both sides have the same value. It is false when they do not.

For example:

• \(4 + 6 = 10\) is true.
• \(9 = 3 + 5\) is false, because \(3 + 5 = 8\), not 9.
• \(x + 2 = 7\) is true only when \(x = 5\).

Why this matters

The balance model is one of the most important ideas in algebra. It helps you solve equations in a fair and organized way. As equations get harder, the same idea still works: keep both sides balanced and use inverse operations.

Brief Summary

Equality means both sides of an equation have the same value. The balance model shows that equations stay true only when we do the same operation to both sides. We use the Properties of Equality and inverse operations to isolate the variable and solve equations correctly.

Solving One- and Two-Step Equations means finding the value of a variable that makes an equation true.

An equation is a math sentence with an equals sign, like \(x + 5 = 12\). A variable is a letter that stands for an unknown number.

A helpful way to think about equations is the balance model. Imagine a balance scale. If both sides are equal, the scale stays balanced. If you do the same thing to both sides, it stays balanced.

That idea is the key to solving equations: whatever you do to one side, you must do to the other side.

When solving equations, we use inverse operations. Inverse operations undo each other:

• Addition and subtraction are inverses.
• Multiplication and division are inverses.

Our goal is to isolate the variable, which means get the variable by itself on one side of the equation.

One-step equations need only one inverse operation to solve. Two-step equations need two inverse operations.

For two-step equations, work in reverse order of operations. Since order of operations says multiply/divide before add/subtract, solving usually goes the other way: undo addition or subtraction first, then undo multiplication or division.

Main idea: Keep the equation balanced and undo operations step by step until the variable is alone.

How to solve a one-step equation

1. Look at what is happening to the variable.
2. Use the inverse operation to undo it.
3. Do the same thing to both sides.
4. Check your answer by substituting it back into the equation.

Worked Example 1: One-step addition equation

Solve \(x + 7 = 19\).

The variable \(x\) has 7 added to it. To undo that, subtract 7 from both sides.

Check:

This is true, so the solution is \(x = 12\).

Worked Example 2: One-step multiplication equation

Solve \(5y = 35\).

The variable \(y\) is being multiplied by 5. To undo that, divide both sides by 5.

Check:

This is true, so the solution is \(y = 7\).

How to solve a two-step equation

1. Find the operations attached to the variable.
2. Undo the addition or subtraction first.
3. Then undo the multiplication or division.
4. Do each step to both sides.
5. Check your answer.

Worked Example 3: Two-step equation with addition and multiplication

Solve \(3x + 4 = 19\).

Here, \(x\) is multiplied by 3, then 4 is added. Undo the 4 first by subtracting 4 from both sides.

Now undo the multiplication by 3 by dividing both sides by 3.

Check:

This is true, so the solution is \(x = 5\).

Worked Example 4: Two-step equation with subtraction and division

Solve $$\frac{m}{4} - 2 = 6$$

The variable \(m\) is divided by 4, then 2 is subtracted. Undo the subtraction first by adding 2 to both sides.

Now undo the division by 4 by multiplying both sides by 4.

Check:

This is true, so the solution is \(m = 32\).

Tips for success

• Keep the equation balanced: Always do the same operation to both sides.
• Use inverse operations: Subtract to undo addition, add to undo subtraction, divide to undo multiplication, and multiply to undo division.
• Work step by step: Do not try to do too much at once.
• Solve in reverse order: For two-step equations, undo the last thing done to the variable first.
• Check your answer: Put the value back into the original equation.

Common mistakes to avoid

• Only changing one side of the equation.
• Using the wrong inverse operation.
• Not following reverse order when solving two-step equations.
• Forgetting to check the answer.

Quick review

If you see \(x + 9 = 14\), subtract 9 from both sides.

If you see \(6x = 42\), divide both sides by 6.

If you see \(2x - 3 = 11\), add 3 first, then divide by 2.

If you see $$\frac{x}{5} + 1 = 7$$, subtract 1 first, then multiply by 5.

Summary

To solve one- and two-step equations, isolate the variable by using inverse operations. Keep the equation balanced by doing the same thing to both sides. For two-step equations, undo operations in reverse order, and always check your solution at the end.

Equations with Rational Coefficients

Sometimes an equation has a number in front of the variable that is a fraction or a decimal. These are called rational coefficients.

For example, in the equation \(\frac{3}{4}x = 12\), the coefficient of \(x\) is \(\frac{3}{4}\). In the equation \(0.6y = 9\), the coefficient of \(y\) is \(0.6\).

The goal is still the same as in all one-step and two-step equations: isolate the variable. That means get the variable by itself.

A helpful way to think about solving equations is the balance model. An equation is like a balanced scale. Whatever you do to one side, you must also do to the other side to keep it balanced.

There are two common ways to solve equations with rational coefficients:

• Use the reciprocal to undo a fraction coefficient.
• Clear the denominators by multiplying every term by the same number.

Let’s learn each method.

1. Using reciprocals

A reciprocal is the “flip” of a fraction. The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). When you multiply a fraction by its reciprocal, the result is \(1\).

For example,

That is useful because multiplying by \(1\) leaves just the variable:

So if a variable has a fraction coefficient, you can multiply both sides of the equation by the reciprocal of that fraction.

2. Clearing denominators

Sometimes an equation has fractions in more than one place, such as

In a case like this, it can be easier to get rid of all the fractions first. To do that, multiply every term by a common denominator.

The denominators here are \(2\), \(4\), and \(2\). A common denominator is \(4\). Multiplying every term by \(4\) removes the fractions and gives an easier equation to solve.

3. What about decimals?

Decimals are also rational numbers. You can solve decimal equations by dividing or multiplying, just like with whole numbers.

Sometimes it helps to turn the decimal into a fraction. For example, \(0.5 = \frac{1}{2}\) and \(1.2 = \frac{12}{10} = \frac{6}{5}\). Other times, it is easier to keep the decimal and use inverse operations.

Important solving steps

1. Look at the equation carefully.
2. Decide whether using a reciprocal or clearing denominators will be easier.
3. Do the same operation to both sides.
4. Simplify step by step.
5. Check your answer by substituting it back into the original equation.

Worked Example 1: One-step equation with a fraction coefficient

Solve:

The coefficient of \(x\) is \(\frac{2}{5}\). Its reciprocal is \(\frac{5}{2}\).

Multiply both sides by \(\frac{5}{2}\):

Simplify:

Check:

The answer is correct, so

Worked Example 2: One-step equation with a decimal coefficient

Solve:

To undo multiplying by \(0.4\), divide both sides by \(0.4\):

Check:

So the solution is

Worked Example 3: Two-step equation with fractions

Solve:

First, subtract \(2\) from both sides:

Now undo \(\frac{3}{4}\) by multiplying both sides by its reciprocal, \(\frac{4}{3}\):

Check:

So,

Worked Example 4: Clearing denominators

Solve:

The denominators are \(3\), \(6\), and \(6\). A common denominator is \(6\).

Multiply every term by \(6\):

Simplify each term:

Now solve the simpler equation.

Subtract \(1\) from both sides:

Divide both sides by \(2\):

Check:

That means we need to look back. Let’s check the solving steps again.

Starting from

Multiply every term by \(6\):

Subtract \(1\):

Divide by \(2\):

Now check carefully in the original equation:

Write \(\frac{2}{3}\) as sixths:

So,

The answer is correct. Therefore,

Tips for success

• If the coefficient is a fraction, think: multiply by the reciprocal.
• If there are several fractions, think: clear the denominators.
• If the coefficient is a decimal, you can often divide by that decimal.
• Always do the same thing to both sides of the equation.
• Always check your solution.

Common mistakes to avoid

• Multiplying only one term when clearing denominators. You must multiply every term.
• Forgetting to use the reciprocal of a fraction coefficient.
• Making errors when adding fractions during the check.
• Stopping too early before the variable is by itself.

Quick practice ideas

• Solve \(\frac{5}{6}x = 10\)
• Solve \(1.5m = 12\)
• Solve \(\frac{2}{3}n - 1 = 7\)
• Solve \(\frac{1}{4}p + \frac{3}{8} = \frac{7}{8}\)

Summary

Equations with rational coefficients can be solved using the same balance idea as other equations. When the coefficient is a fraction, multiply both sides by the reciprocal. When an equation has several fractions, multiply every term by a common denominator to clear the fractions. When the coefficient is a decimal, use inverse operations like division. Then always check your answer in the original equation.

Lesson: Multi-Step Equations and Combining Terms

When you solve an equation, your goal is to find the value of the variable that makes the equation true. In this lesson, you will learn how to solve multi-step equations, especially equations that require you to distribute and combine like terms before solving.

These equations may look more complicated at first, but they follow the same big idea as simpler equations: keep the equation balanced and undo operations step by step.

What is a multi-step equation?

A multi-step equation is an equation that takes more than one step to solve. You may need to:

• Use the distributive property,
• Combine like terms,
• Add or subtract terms from both sides,
• Multiply or divide to isolate the variable.

What are like terms?

Like terms are terms that have the same variable part. For example, \(3x\) and \(-5x\) are like terms because they both have \(x\). Also, \(7\) and \(-2\) are like terms because they are both constants.

These are like terms:

• \(4x\) and \(9x\)
• \(-2y\) and \(5y\)
• \(8\) and \(-3\)

These are not like terms:

• \(3x\) and \(3y\)
• \(4x\) and \(4x^2\)
• \(2a\) and \(2\)

How do you combine like terms?

To combine like terms, add or subtract their coefficients.

Example:

$$3x + 5x = 8x$$

$$10 - 4 = 6$$

$$7y - 2y + 3 = 5y + 3$$

What is the distributive property?

The distributive property means multiplying a number outside parentheses by every term inside the parentheses.

$$a(b + c) = ab + ac$$

For example:

$$3(x + 4) = 3x + 12$$

$$-2(x - 5) = -2x + 10$$

Be very careful with negative signs. A negative number distributed across parentheses can change the signs of the terms inside.

Steps for solving multi-step equations

1. Use the distributive property if there are parentheses.
2. Combine like terms on each side of the equation.
3. Move variable terms to one side and constants to the other side.
4. Use inverse operations to isolate the variable.
5. Check your answer by substituting it back into the original equation.

Think of solving equations like cleaning up a messy room. First, simplify what you can. Then solve what is left.

Worked Example 1: Combine like terms first

Solve:

$$3x + 4 + 2x = 19$$

Step 1: Combine like terms.

\(3x\) and \(2x\) are like terms.

$$5x + 4 = 19$$

Step 2: Subtract 4 from both sides.

$$5x = 15$$

Step 3: Divide both sides by 5.

$$x = 3$$

Check:

Substitute \(x = 3\) into the original equation:

$$3(3) + 4 + 2(3) = 9 + 4 + 6 = 19$$

The answer is correct.

Worked Example 2: Use distribution

Solve:

$$4(x + 2) = 20$$

Step 1: Distribute the 4.

$$4x + 8 = 20$$

Step 2: Subtract 8 from both sides.

$$4x = 12$$

Step 3: Divide both sides by 4.

$$x = 3$$

Check:

$$4(3 + 2) = 4(5) = 20$$

The answer is correct.

Worked Example 3: Distribute and combine like terms

Solve:

$$2(x + 3) + 4x = 18$$

Step 1: Distribute.

$$2x + 6 + 4x = 18$$

Step 2: Combine like terms.

\(2x + 4x = 6x\)

$$6x + 6 = 18$$

Step 3: Subtract 6 from both sides.

$$6x = 12$$

Step 4: Divide by 6.

$$x = 2$$

Check:

$$2(2 + 3) + 4(2) = 2(5) + 8 = 10 + 8 = 18$$

The answer is correct.

Worked Example 4: Variables on both sides

Solve:

$$3(x + 1) + 2x = x + 14$$

Step 1: Distribute.

$$3x + 3 + 2x = x + 14$$

Step 2: Combine like terms on the left side.

$$5x + 3 = x + 14$$

Step 3: Subtract \(x\) from both sides.

$$4x + 3 = 14$$

Step 4: Subtract 3 from both sides.

$$4x = 11$$

Step 5: Divide by 4.

$$x = \frac{11}{4}$$

You can also write this as \(2\frac{3}{4}\).

Check:

Substitute \(x = \frac{11}{4}\) into the original equation.

Left side:

$$3\left(\frac{11}{4} + 1\right) + 2\left(\frac{11}{4}\right)$$

$$3\left(\frac{15}{4}\right) + \frac{22}{4} = \frac{45}{4} + \frac{22}{4} = \frac{67}{4}$$

Right side:

$$\frac{11}{4} + 14 = \frac{11}{4} + \frac{56}{4} = \frac{67}{4}$$

Both sides are equal, so the answer is correct.

Tips for success

• Distribute carefully. Multiply the number outside the parentheses by every term inside.
• Watch negative signs. They can change the signs of terms when you distribute.
• Combine only like terms. Terms must have the same variable part.
• Keep the equation balanced. Whatever you do to one side, do to the other side.
• Check your solution. Put your answer back into the original equation.

Common mistakes to avoid

• Forgetting to distribute to every term inside parentheses.
• Combining terms that are not like terms.
• Making sign mistakes with subtraction or negative numbers.
• Solving before simplifying the equation.

Quick practice ideas

Try solving these on your own:

• \(5(x + 1) = 25\)
• \(2x + 7 + 3x = 22\)
• \(3(x - 2) + x = 10\)
• \(4(x + 3) = 2x + 18\)

Summary

Multi-step equations often need you to simplify before solving. Start by using the distributive property, then combine like terms. After that, use inverse operations to isolate the variable. If you work carefully and keep both sides balanced, you can solve even complicated equations step by step.

Variables on Both Sides means solving an equation where the variable appears on the left side and the right side of the equals sign.

For example, in the equation \(3x + 5 = x + 17\), the variable \(x\) is on both sides. Our job is to move the variable terms to one side, move the number terms to the other side, and then solve.

A helpful way to think about equations is the balance model. The equals sign means both sides are equal, like two sides of a balanced scale. If you do the same thing to both sides, the equation stays balanced.

The goal: get all the variable terms on one side and all the constant terms on the other side.

Here is the basic plan:

1. Choose which side will keep the variable. Usually, it is easiest to keep the variable positive.
2. Add or subtract to move variable terms to one side.
3. Add or subtract to move number terms to the other side.
4. Divide or multiply if needed to get the variable alone.
5. Check your answer by substituting it back into the original equation.

Important words:

• Variable: a letter that stands for an unknown number, like \(x\).
• Constant: a number without a variable, like \(7\) or \(-3\).
• Coefficient: the number in front of a variable, like the \(4\) in \(4x\).

When solving equations with variables on both sides, it helps to group like terms carefully:

• Variable terms go together: \(3x\), \(-2x\), \(x\)
• Constant terms go together: \(5\), \(-8\), \(12\)

Let’s look at some examples.

Example 1: A basic equation with variables on both sides

Solve:

$$3x + 5 = x + 17$$

Step 1: Move the variable terms to one side. Subtract \(x\) from both sides.

$$3x - x + 5 = x - x + 17$$

$$2x + 5 = 17$$

Step 2: Move the constant term away from the variable. Subtract \(5\) from both sides.

$$2x + 5 - 5 = 17 - 5$$

$$2x = 12$$

Step 3: Divide both sides by \(2\).

$$x = 6$$

Check:

Left side: \(3(6) + 5 = 18 + 5 = 23\)

Right side: \(6 + 17 = 23\)

Both sides are equal, so \(x = 6\) is correct.

Example 2: Keep the variable positive if possible

Solve:

$$2x + 9 = 5x - 6$$

Here, we can move the smaller variable term, \(2x\), to the right by subtracting \(2x\) from both sides.

$$2x - 2x + 9 = 5x - 2x - 6$$

$$9 = 3x - 6$$

Now move the constant term. Add \(6\) to both sides.

$$9 + 6 = 3x - 6 + 6$$

$$15 = 3x$$

Divide both sides by \(3\).

$$5 = x$$

So,

$$x = 5$$

Check:

Left side: \(2(5) + 9 = 10 + 9 = 19\)

Right side: \(5(5) - 6 = 25 - 6 = 19\)

The answer is correct.

Example 3: Variables and negative numbers

Solve:

$$4x - 7 = 2x + 9$$

Subtract \(2x\) from both sides to move variables to one side.

$$4x - 2x - 7 = 2x - 2x + 9$$

$$2x - 7 = 9$$

Add \(7\) to both sides.

$$2x - 7 + 7 = 9 + 7$$

$$2x = 16$$

Divide by \(2\).

$$x = 8$$

Check:

Left side: \(4(8) - 7 = 32 - 7 = 25\)

Right side: \(2(8) + 9 = 16 + 9 = 25\)

Correct.

Example 4: Variables on both sides with distribution first

Sometimes you must simplify before moving terms.

Solve:

$$2(x + 3) = x + 11$$

Step 1: Use the distributive property.

$$2x + 6 = x + 11$$

Step 2: Subtract \(x\) from both sides.

$$2x - x + 6 = x - x + 11$$

$$x + 6 = 11$$

Step 3: Subtract \(6\) from both sides.

$$x = 5$$

Check:

Original equation: \(2(x + 3) = x + 11\)

Left side: \(2(5 + 3) = 2(8) = 16\)

Right side: \(5 + 11 = 16\)

So \(x = 5\) is correct.

Tips for success

• Do the same thing to both sides. This keeps the equation balanced.
• Move variable terms together. Add or subtract to collect them on one side.
• Move constants together. Put plain numbers on the other side.
• Work one step at a time. Don’t try to do too much at once.
• Watch negative signs carefully. A small sign mistake can change the answer.
• Always check your answer. Substitute your value back into the original equation.

What if the variable disappears?

Sometimes, after moving variables, you may get an equation like:

$$5 = 5$$

This means the equation is true for all values of the variable.

Other times, you may get something like:

$$5 = 9$$

This is never true, so the equation has no solution.

Here is a quick example of each:

• \(2x + 4 = 2x + 4\) leads to \(4 = 4\), so there are infinitely many solutions.
• \(3x + 1 = 3x + 8\) leads to \(1 = 8\), so there is no solution.

Common mistakes to avoid

• Forgetting to do the same operation to both sides.
• Combining unlike terms, such as trying to combine \(3x\) and \(5\).
• Making sign errors with negatives.
• Stopping too early before the variable is fully isolated.
• Not checking the answer in the original equation.

Let’s review the big idea.

To solve equations with variables on both sides, first collect the variable terms on one side and the constants on the other. Then use inverse operations to get the variable alone. Finally, check your solution to make sure both sides are equal.

Special Case Solutions happen when an equation does not have just one answer. Sometimes an equation is true for every value of the variable, and sometimes it is never true at all.

In this lesson, you will learn how to recognize these special cases when solving linear equations. This is important because not every equation ends with one number like \(x=5\).

Most of the time, when we solve an equation, we get one solution. For example, in \(x+3=7\), we subtract 3 from both sides and get \(x=4\).

But some equations behave differently after we simplify both sides. These are called special case solutions.

1. The three possible results when solving an equation

• One solution: the equation is true for only one value of the variable.
• Infinite solutions: the equation is true for every value of the variable.
• No solution: the equation is not true for any value of the variable.

Let’s look more closely at the last two, because they are the special cases.

2. Infinite solutions

An equation has infinite solutions when both sides of the equation are really the same expression.

After you simplify, the variable terms may cancel out and leave a true statement, such as:

$$5=5$$

or

$$0=0$$

This means the equation is always true, no matter what number you choose for the variable.

Another name for this kind of equation is an identity. For 7th Grade, you can think of an identity as an equation that is always true.

3. No solution

An equation has no solution when the variable terms cancel out but the numbers left behind make a false statement, such as:

$$5=8$$

or

$$0=3$$

These statements are never true. That means there is no value of the variable that can make the original equation true.

Another name for this kind of equation is a contradiction. For 7th Grade, you can think of a contradiction as an equation that is never true.

4. How to tell which special case you have

When solving a linear equation, follow the usual steps:

1. Simplify both sides if needed.
2. Use inverse operations to move variable terms and number terms.
3. Watch carefully for what happens when the variable disappears.

If the variable disappears and you get a true statement, the equation has infinite solutions.

If the variable disappears and you get a false statement, the equation has no solution.

If the variable does not disappear and you end with something like \(x=2\), then the equation has one solution.

5. Why this happens

Special cases often happen when there are variable terms on both sides of the equation.

If both sides match perfectly after simplifying, then every value works. If the variable parts match but the number parts do not, then nothing works.

Here is the idea:

• Same variable part and same number part \(\) infinite solutions
• Same variable part but different number part \(\) no solution

6. Worked examples

Example 1: Infinite solutions

Solve:

$$3x+4=x+2x+4$$

Step 1: Simplify both sides.

On the right side, \(x+2x=3x\). So the equation becomes:

$$3x+4=3x+4$$

Step 2: Subtract \(3x\) from both sides.

$$4=4$$

Step 3: Interpret the result.

The statement \(4=4\) is always true.

So this equation has infinite solutions.

Answer: Infinite solutions.

Example 2: No solution

Solve:

$$2x+7=2x+1$$

Step 1: Subtract \(2x\) from both sides.

$$7=1$$

Step 2: Interpret the result.

The statement \(7=1\) is false.

There is no value of \(x\) that can make the equation true.

Answer: No solution.

Example 3: One solution

Solve:

$$4x-5=11$$

Step 1: Add 5 to both sides.

$$4x=16$$

Step 2: Divide both sides by 4.

$$x=4$$

This equation has one solution.

Answer: \(x=4\)

This example is not a special case, but it helps you compare normal equations with special-case equations.

Example 4: Special case with parentheses

Solve:

$$3(x+2)=3x+9$$

Step 1: Distribute on the left side.

$$3x+6=3x+9$$

Step 2: Subtract \(3x\) from both sides.

$$6=9$$

Step 3: Interpret the result.

The statement \(6=9\) is false.

So the equation has no solution.

Answer: No solution.

7. Quick ways to check your thinking

Ask yourself these questions after solving:

• Did the variable disappear completely?
• If yes, is the result true or false?
• True statement \(\) infinite solutions
• False statement \(\) no solution

You can also test a value to make sense of the answer.

For an equation with infinite solutions, try any number for the variable. Both sides should match.

For example, in \(3x+4=x+2x+4\):

• If \(x=1\), then left side \(=3(1)+4=7\), right side \(=1+2(1)+4=7\)
• If \(x=5\), then left side \(=3(5)+4=19\), right side \(=5+10+4=19\)

Every value works.

For an equation with no solution, no matter what value you try, the two sides will not match.

For example, in \(2x+7=2x+1\):

• If \(x=0\), then left side \(=7\), right side \(=1\)
• If \(x=3\), then left side \(=13\), right side \(=7\)

The sides are never equal.

8. Common mistakes to avoid

• Mistake: Thinking \(4=4\) means the answer is \(x=4\).
  Fix: \(4=4\) is a true statement, so the answer is infinite solutions, not \(x=4\).
• Mistake: Thinking \(7=1\) means \(x=1\).
  Fix: \(7=1\) is false, so the answer is no solution.
• Mistake: Forgetting to simplify both sides first.
  Fix: Combine like terms and distribute before deciding what kind of solution you have.

9. Practice thinking

Try to decide what type of answer each equation has:

• \(5x+2=5x+2\)
• \(7x-1=7x+6\)
• \(2x+8=18\)

Answers:

• \(5x+2=5x+2\) \(\) infinite solutions
• \(7x-1=7x+6\) \(\) no solution
• \(2x+8=18\) \(\) one solution

10. Summary

When solving linear equations, you may get one solution, infinite solutions, or no solution.

If the variable disappears and you get a true statement like \(0=0\) or \(4=4\), the equation has infinite solutions.

If the variable disappears and you get a false statement like \(0=5\) or \(6=9\), the equation has no solution.

Always simplify carefully, then look at the final statement to decide what kind of solution the equation has.

Literal Equations and Formulas

In math, a literal equation is an equation that has more than one variable. Many formulas you already know are literal equations.

For example, the formula for the area of a rectangle is \(A = lw\). This equation has three variables: \(A\), \(l\), and \(w\).

Sometimes you are given a formula and asked to solve for a different variable. This means you need to rearrange the equation so the requested variable is alone on one side.

This is just like solving a regular equation. You use the balance model: whatever you do to one side, you must do to the other side.

Why this matters: formulas are used in geometry, science, and everyday life. If you know how to rearrange a formula, you can find missing information in many situations.

Main idea: To solve a literal equation, isolate the variable you want by using inverse operations.

• Add and subtract are inverse operations.
• Multiply and divide are inverse operations.

Steps for solving a literal equation

1. Find the variable you are asked to solve for.
2. Look at what is being done to that variable.
3. Use inverse operations to undo those operations.
4. Do the same thing to both sides of the equation.
5. Check that the variable is alone.

Important note: In formulas, letters usually stand for numbers. They are variables, not labels. Your job is to move them the same way you would move numbers in an equation.

Example 1: Solve for a variable being multiplied

Formula: \(A = lw\)

Solve for \(w\).

In this formula, \(w\) is being multiplied by \(l\).

To undo multiplication by \(l\), divide both sides by \(l\).

So the formula solved for \(w\) is:

This means if you know the area and the length, you can find the width.

Example 2: Solve for a variable with addition

Formula: \(P = 2l + 2w\)

Solve for \(w\).

First, move the part that does not include \(w\). The term \(2l\) is added, so subtract \(2l\) from both sides.

Now \(w\) is still being multiplied by 2, so divide both sides by 2.

So the formula solved for \(w\) is:

Tip: When there is more than one step attached to the variable, undo them one at a time.

Example 3: Solve for a variable in a temperature formula

Formula: \(F = \frac{9}{5}C + 32\)

Solve for \(C\).

First, subtract 32 from both sides because 32 is added to the \(C\)-part.

Now \(C\) is multiplied by \(\frac{9}{5}\). To undo that, multiply both sides by the reciprocal, \(\frac{5}{9}\).

So the formula solved for \(C\) is:

This is the formula used to convert Fahrenheit to Celsius.

Example 4: Solve for a variable in a formula with division

Formula: \(d = rt\)

Solve for \(t\).

Here, \(t\) is multiplied by \(r\). Divide both sides by \(r\).

So the formula solved for \(t\) is:

This formula can help you find time if you know distance and rate.

What to watch out for

• Do not move letters by themselves unless you use an operation on both sides.
• Undo addition or subtraction first, then undo multiplication or division.
• Be careful with fractions. Multiplying by the reciprocal can help.
• Keep the variable you want on one side alone.

Quick check

Try thinking through these:

• If \(A = bh\), then solving for \(h\) gives \(h = \frac{A}{b}\).
• If \(y = mx + b\), then solving for \(x\) gives \(x = \frac{y - b}{m}\).

How to check your answer

One good way to check is to look back at the original formula and make sure your new equation is reasonable.

For example, if \(A = lw\), then solving for \(w\) as \(w = \frac{A}{l}\) makes sense because width should equal area divided by length.

Summary

Literal equations are formulas with more than one variable. To solve for a specific variable, use inverse operations and keep the equation balanced.

Start by removing terms added or subtracted, then undo multiplication or division. With practice, you can rearrange formulas to find any variable you need.

Inequality Notation and Constraints

In math, we do not always need an exact answer. Sometimes we only know that a value must be more than, less than, at least, or at most a certain number. In these cases, we use inequalities.

An inequality compares two amounts that are not always equal. Inequalities are very useful for describing constraints, which are limits or rules a situation must follow.

For example, if a bus can hold no more than 40 students, the number of students is not exactly 40 every time. It could be 25, 32, or 40. This kind of rule is best shown with an inequality.

Inequality Symbols

Here are the main symbols you need to know:

• Less than: \( < \)
• Greater than: \( > \)
• Less than or equal to: \( \leq \)
• Greater than or equal to: \( \geq \)

These symbols help us describe number ranges.

• \(x < 5\) means \(x\) is any number less than 5.
• \(x > 5\) means \(x\) is any number greater than 5.
• \(x \leq 5\) means \(x\) is less than 5 or equal to 5.
• \(x \geq 5\) means \(x\) is greater than 5 or equal to 5.

Understanding Key Words

Word problems often include clue words that tell you which inequality symbol to use.

• Less than means \( < \)
• More than or greater than means \( > \)
• At most means \( \leq \)
• No more than means \( \leq \)
• At least means \( \geq \)
• No less than means \( \geq \)

It is very important to notice whether the number itself is included.

• If the value can equal the number, use \(\leq\) or \(\geq\).
• If the value cannot equal the number, use \(<\) or \(>\).

What Is a Constraint?

A constraint is a limit or condition in a real-world problem. It tells what values are allowed.

For example:

• A ride may require a height of at least 48 inches.
• A backpack may hold no more than 12 pounds.
• You may spend less than 20 dollars.

Each of these statements can be written as an inequality.

Writing Inequalities from Real Situations

To write an inequality, follow these steps:

1. Choose a variable for the unknown amount.
2. Decide what the situation is saying about that amount.
3. Pick the correct inequality symbol.
4. Write the inequality.

Worked Example 1: Less Than

A student wants to buy a game that costs less than 30 dollars. Let \(p\) be the price of the game.

The words less than 30 dollars mean the price must be below 30, but not equal to 30.

So the inequality is:

This means any price under 30 dollars will work.

Worked Example 2: At Most

A box can weigh at most 15 pounds. Let \(w\) be the weight of the box.

The phrase at most 15 pounds means the weight can be 15 or any number less than 15.

So the inequality is:

This includes 15, because at most means the number is allowed to equal 15.

Worked Example 3: At Least

You must score at least 70 points to pass a test. Let \(s\) be the test score.

The words at least 70 mean 70 or more.

So the inequality is:

This means 70, 71, 85, and 100 are all possible passing scores.

Worked Example 4: Combining a Real-World Constraint with a Number Line

A theater has 120 seats. The number of tickets sold cannot be more than 120. Let \(t\) be the number of tickets sold.

The phrase cannot be more than 120 means no more than 120.

So the inequality is:

To graph this on a number line:

• Put a closed circle at 120, because 120 is included.
• Shade to the left, because numbers less than 120 also work.

Graphing Inequalities on a Number Line

A number line helps you show all possible solutions to an inequality.

There are two important parts:

• Open circle: use this when the endpoint is not included. This happens with \(<\) or \(>\).
• Closed circle: use this when the endpoint is included. This happens with \(\leq\) or \(\geq\).

You also need to know which direction to shade:

• Shade left for numbers less than a value.
• Shade right for numbers greater than a value.

Examples:

• \(x < 4\): open circle at 4, shade left
• \(x \leq 4\): closed circle at 4, shade left
• \(x > 4\): open circle at 4, shade right
• \(x \geq 4\): closed circle at 4, shade right

Matching Words, Symbols, and Graphs

Here is a helpful way to connect them:

• More than 8 \(\rightarrow x > 8\) \(\rightarrow\) open circle at 8, shade right
• At least 8 \(\rightarrow x \geq 8\) \(\rightarrow\) closed circle at 8, shade right
• Less than 8 \(\rightarrow x < 8\) \(\rightarrow\) open circle at 8, shade left
• At most 8 \(\rightarrow x \leq 8\) \(\rightarrow\) closed circle at 8, shade left

Common Mistakes to Avoid

• Mixing up \(<\) and \(\leq\). Ask yourself: is the number included?
• Mixing up \(>\) and \(\geq\). Again, check whether the number can be equal.
• Shading the wrong direction on a number line. Less than goes left. Greater than goes right.
• Using the wrong clue words. For example, at least does not mean less than. It means greater than or equal to.

Quick Practice Thinking

Try to decide which inequality matches each statement:

• The temperature is no more than 10 degrees: \(t \leq 10\)
• A runner trains more than 5 miles: \(m > 5\)
• You need at least 3 tickets: \(n \geq 3\)
• A jar holds less than 2 liters: \(j < 2\)

Why Inequalities Matter

Inequalities help us describe real-life limits. We use them when talking about budgets, time limits, weight limits, height requirements, scores, and many other situations.

Instead of showing just one answer, inequalities show a whole set of possible answers that follow the rule.

Summary

An inequality compares values that may not be equal. The symbols \(<\), \(>\), \(\leq\), and \(\geq\) help describe limits and constraints in real-world situations.

When writing an inequality, pay close attention to clue words like less than, more than, at most, and at least. On a number line, use an open circle if the endpoint is not included and a closed circle if it is included.

Solving Multi-Step Inequalities

Sometimes in math, we do not need an exact value. Instead, we need to know which values are greater than, less than, at least, or at most a number. That is what inequalities help us show.

A multi-step inequality is an inequality that takes more than one step to solve. You may need to combine like terms, use the distributive property, and then use inverse operations to get the variable by itself.

Solving inequalities is a lot like solving equations, but there is one very important extra rule: if you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

Inequality symbols

•  means less than
•  means greater than
• = means less than or equal to
• = means greater than or equal to

For example:

• \(x > 4\) means \(x\) is any number greater than 4.
• \(x \le 7\) means \(x\) is any number less than or equal to 7.

How solving inequalities is like solving equations

You can do the same operation to both sides of an inequality and keep it balanced.

• Add the same number to both sides.
• Subtract the same number from both sides.
• Multiply both sides by the same positive number.
• Divide both sides by the same positive number.

These actions keep the inequality true.

The special rule

If you multiply or divide both sides by a negative number, the inequality symbol must flip.

For example:

Start with

$$-2x > 10$$

Divide both sides by \(-2\):

$$x < -5$$

The symbol changed from \(>\) to \(<\) because we divided by a negative number.

Why does the sign flip?

Think about these numbers:

$$3 < 5$$

Now multiply both sides by \(-1\):

$$-3 > -5$$

The order changes, so the symbol must reverse.

Steps for solving multi-step inequalities

1. Simplify each side if needed.
2. Use the distributive property if needed.
3. Combine like terms.
4. Use inverse operations to get the variable alone.
5. If you multiply or divide by a negative number, reverse the inequality sign.
6. Check your answer by testing a value.

Worked Example 1: Two-step inequality

Solve:

$$x + 5 > 12$$

Subtract 5 from both sides:

$$x > 7$$

Answer: \(x > 7\)

Check with a number greater than 7, like 8:

$$8 + 5 = 13$$

Since \(13 > 12\), the solution makes sense.

Worked Example 2: More than one step

Solve:

$$3x - 4 \le 11$$

Add 4 to both sides:

$$3x \le 15$$

Divide both sides by 3:

$$x \le 5$$

We do not reverse the sign because we divided by a positive number.

Answer: \(x \le 5\)

Check with \(x = 5\):

$$3(5) - 4 = 15 - 4 = 11$$

Since \(11 \le 11\), the answer is correct.

Worked Example 3: Distributive property

Solve:

$$2(x + 3) > 14$$

Distribute the 2:

$$2x + 6 > 14$$

Subtract 6 from both sides:

$$2x > 8$$

Divide both sides by 2:

$$x > 4$$

Answer: \(x > 4\)

Check with \(x = 5\):

$$2(5 + 3) = 2(8) = 16$$

Since \(16 > 14\), the solution works.

Worked Example 4: Dividing by a negative number

Solve:

$$-3x + 2 \ge 14$$

Subtract 2 from both sides:

$$-3x \ge 12$$

Now divide both sides by \(-3\). Since we are dividing by a negative number, we must reverse the inequality sign:

$$x \le -4$$

Answer: \(x \le -4\)

Check with \(x = -4\):

$$-3(-4) + 2 = 12 + 2 = 14$$

Since \(14 \ge 14\), the solution is correct.

Example with variables on both sides

Solve:

$$4x - 1 < 2x + 9$$

Subtract \(2x\) from both sides:

$$2x - 1 < 9$$

Add 1 to both sides:

$$2x < 10$$

Divide both sides by 2:

$$x < 5$$

Answer: \(x < 5\)

Writing solutions in words

• \(x > 7\): all numbers greater than 7
• \(x \le 5\): all numbers less than or equal to 5
• \(x \lt 5\): all numbers less than 5
• \(x \ge -4\): all numbers greater than or equal to \(-4\)

Common mistakes to avoid

• Forgetting to flip the sign when multiplying or dividing by a negative number.
• Not distributing correctly in expressions like \(3(x - 2)\).
• Combining unlike terms, such as trying to combine \(2x\) and 3.
• Only doing an operation on one side instead of both sides.

Quick tip

If you see a negative number attached to the variable right before your last step, pause and ask: Am I dividing by a negative? If the answer is yes, flip the inequality sign.

Practice ideas

Try solving these on your own:

• \(x - 7 < 2\)
• \(5x + 1 \ge 16\)
• \(3(x + 2) \le 18\)
• \(-2x + 4 > 10\)

Summary

To solve a multi-step inequality, simplify both sides, use inverse operations, and isolate the variable. The most important rule to remember is this: when you multiply or divide both sides by a negative number, reverse the inequality symbol.

Just like equations, inequalities can be checked by substituting a value that should work. If you move carefully step by step, you can solve even more complicated inequalities with confidence.

Compound Inequalities are two inequalities joined together in one statement. They help us describe a set of numbers that make both parts true or at least one part true.

In 7th Grade, there are two main kinds of compound inequalities:

• AND inequalities: both parts must be true at the same time.
• OR inequalities: at least one part must be true.

Learning compound inequalities means learning how to solve them and how to graph their solutions on a number line.

1. What is an inequality?

An inequality compares two values. Instead of saying two sides are equal, it shows that one side is greater or less than the other.

• \(<\) means less than
• \(>\) means greater than
• \(\le\) means less than or equal to
• \(\ge\) means greater than or equal to

For example, in \(x < 5\), any number less than 5 is a solution.

2. Understanding AND inequalities

An AND compound inequality means the solution must satisfy both inequalities. This usually creates a range of values between two numbers.

Example of an AND inequality:

\(2 < x \le 7\)

This means:

• \(x\) is greater than 2
• and \(x\) is less than or equal to 7

So the solutions are all numbers between 2 and 7, not including 2, but including 7.

On a number line:

• Use an open circle at 2 because 2 is not included.
• Use a closed circle at 7 because 7 is included.
• Shade the part between 2 and 7.

3. Understanding OR inequalities

An OR compound inequality means the solution must satisfy at least one of the inequalities. This usually creates two separate groups of numbers.

Example of an OR inequality:

\(x < -1\) or \(x \ge 4\)

This means:

• all numbers less than \(-1\)
• and also all numbers greater than or equal to 4

On a number line:

• Use an open circle at \(-1\) and shade left.
• Use a closed circle at 4 and shade right.

Important idea:

• AND means the overlap, or intersection.
• OR means both solution groups together, or union.

4. How to solve compound inequalities

When solving compound inequalities, use the same skills you use for regular inequalities. You can add, subtract, multiply, or divide to isolate the variable.

Remember: if you multiply or divide by a negative number, you must flip the inequality sign.

For AND inequalities written in one line, do the same operation to all three parts.

For example:

$$3 < x + 2 \le 8$$

Subtract 2 from all three parts:

$$3 - 2 < x + 2 - 2 \le 8 - 2$$

$$1 < x \le 6$$

So the solution is all numbers greater than 1 and less than or equal to 6.

5. Worked Examples

Example 1: Solve an AND inequality

Solve:

$$-4 \le x + 1 < 6$$

Subtract 1 from all three parts:

$$-4 - 1 \le x + 1 - 1 < 6 - 1$$

$$-5 \le x < 5$$

Answer: \(x\) can be any number from \(-5\) up to 5, including \(-5\) but not including 5.

Graph:

• Closed circle at \(-5\)
• Open circle at 5
• Shade between them

Example 2: Solve an OR inequality

Solve:

$$x - 3 < -2 \quad \text{or} \quad x + 1 \ge 7$$

Solve each part separately.

First part:

$$x - 3 < -2$$

Add 3 to both sides:

$$x < 1$$

Second part:

$$x + 1 \ge 7$$

Subtract 1 from both sides:

$$x \ge 6$$

Answer:

$$x < 1 \quad \text{or} \quad x \ge 6$$

Graph:

• Open circle at 1 and shade left
• Closed circle at 6 and shade right

Example 3: Solve an AND inequality with multiplication

Solve:

$$2 \le \frac{x}{3} < 5$$

Multiply all three parts by 3:

$$2(3) \le \frac{x}{3}(3) < 5(3)$$

$$6 \le x < 15$$

Answer: \(x\) is at least 6 and less than 15.

Example 4: Solve when dividing by a negative number

Solve:

$$-2x > 8 \quad \text{or} \quad x - 4 \le 3$$

Solve each part separately.

First part:

$$-2x > 8$$

Divide both sides by \(-2\). Since we divide by a negative, flip the inequality sign:

$$x < -4$$

Second part:

$$x - 4 \le 3$$

Add 4 to both sides:

$$x \le 7$$

Now combine with OR:

$$x < -4 \quad \text{or} \quad x \le 7$$

Notice that every number less than \(-4\) is already included in \(x \le 7\). So the full solution is simply:

$$x \le 7$$

This shows that sometimes one part of an OR inequality already covers the other part.

6. How to graph compound inequalities

After solving, graph the solution on a number line.

Use these rules:

• Open circle for \(<\) or \(>\)
• Closed circle for \(\le\) or \(\ge\)
• AND: shade between the two endpoints if the solutions overlap
• OR: shade in both directions or in two separate parts

7. Common mistakes to avoid

• Mixing up AND and OR
• Forgetting to do the same operation to all parts of an AND inequality
• Forgetting to flip the inequality sign when multiplying or dividing by a negative number
• Using a closed circle when the endpoint is not included
• Shading the wrong part of the number line

8. Quick check for understanding

Ask yourself these questions:

• Does this problem use AND or OR?
• Do I need numbers between two values or in separate groups?
• Did I solve both sides correctly?
• Did I use the correct circles on the number line?

9. Summary

Compound inequalities combine two inequalities in one problem. With AND, the solution must make both parts true, so the answer is usually a section between two numbers. With OR, the solution only needs to make one part true, so the answer is often two separate groups.

To solve them, isolate the variable just like in regular inequalities. Then graph the solution carefully using open or closed circles and shading the correct part of the number line.