Equations and Inequalities

Lesson: The Concept of Equality and Solutions

In maths, an equation is a number sentence that says two amounts are equal. An equation always has an equals sign, like this: \(= \). The equals sign means "has the same value as".

For example, in the equation \(5 + 3 = 8\), the left side and the right side both have the same value. Since both sides equal 8, the equation is true.

Sometimes an equation has a letter, such as \(x\), \(n\), or \(m\). The letter stands for an unknown number. Our job is often to figure out which number makes the equation true.

A number that makes an equation true is called a solution. If the number does not make the equation true, then it is not a solution.

Main Idea 1: Equality means both sides are the same

Think of an equation like a balanced scale. The left side and the right side must match in value. If they match, the equation is true. If they do not match, the equation is false.

Look at these examples:

• \(7 = 7\) is true because both sides are the same.
• \(10 = 6 + 4\) is true because \(6 + 4 = 10\).
• \(9 = 12 - 1\) is false because \(12 - 1 = 11\), not 9.

Main Idea 2: A variable stands for an unknown number

In equations, a letter is called a variable. A variable can stand for different numbers in different equations.

For example, in \(x + 2 = 9\), the variable is \(x\). We want to know which number for \(x\) makes the equation true.

Main Idea 3: Check a solution by substituting

To decide if a number is a solution, replace the variable with that number. This is called substituting. Then calculate both sides and see if they are equal.

Here are the steps:

1. Replace the variable with the given number.
2. Simplify each side.
3. Check whether both sides have the same value.
4. If they are equal, the number is a solution.

Worked Example 1

Is \(4\) a solution to \(x + 3 = 7\)?

Step 1: Substitute \(4\) for \(x\).

$$4 + 3 = 7$$

Step 2: Simplify the left side.

$$7 = 7$$

Step 3: Compare both sides.

Both sides are equal, so \(4\) is a solution.

Worked Example 2

Is \(5\) a solution to \(2n = 12\)?

Step 1: Substitute \(5\) for \(n\).

$$2(5) = 12$$

Step 2: Simplify.

$$10 = 12$$

Step 3: Compare both sides.

Since 10 is not equal to 12, \(5\) is not a solution.

Worked Example 3

Which number is a solution to \(m - 6 = 8\): \(12\), \(14\), or \(16\)?

We check each choice.

For \(m = 12\):

$$12 - 6 = 8$$

$$6 = 8$$

Not true, so \(12\) is not a solution.

For \(m = 14\):

$$14 - 6 = 8$$

$$8 = 8$$

True, so \(14\) is a solution.

For \(m = 16\):

$$16 - 6 = 8$$

$$10 = 8$$

Not true, so \(16\) is not a solution.

So, the solution is \(14\).

Worked Example 4

A ticket for a school event costs \(3\) dollars. The total cost is \(15\) dollars. This can be written as:

$$3t = 15$$

Is \(t = 5\) a solution?

Step 1: Substitute \(5\) for \(t\).

$$3(5) = 15$$

Step 2: Simplify.

$$15 = 15$$

Step 3: Compare both sides.

The equation is true, so \(5\) is a solution.

This example shows that equations can represent real-life situations. A solution is the value that makes the situation work correctly.

How to think about true and false equations

• If both sides have the same value, the equation is true.
• If both sides have different values, the equation is false.
• A solution must make the equation true.

Common Mistakes to Avoid

• Thinking the equals sign means “the answer is.” It really means both sides are the same value.
• Forgetting to substitute carefully. Replace every variable with the given number.
• Not checking both sides. Always simplify and compare.

Try These On Your Own

• Is \(3\) a solution to \(x + 4 = 7\)?
• Is \(6\) a solution to \(2a = 12\)?
• Which number is a solution to \(b - 2 = 5\): \(6\), \(7\), or \(8\)?
• Is \(4\) a solution to \(5c = 25\)?

Quick Check Answers

• Yes, because \(3 + 4 = 7\).
• Yes, because \(2 \times 6 = 12\).
• \(7\), because \(7 - 2 = 5\).
• No, because \(5 \times 4 = 20\), not 25.

Summary

An equation is a statement that two sides are equal. A solution is a number that makes the equation true when you substitute it for the variable. To check a solution, replace the variable, simplify, and see whether both sides are equal.

Solving One-Step Equations

Sometimes in math, we do not know the value of a number. We can use a variable, such as \(x\) or \(n\), to stand for that unknown number.

An equation is a math sentence that shows two sides are equal. For example, \(x + 5 = 12\) is an equation.

When we solve an equation, we find the value of the variable that makes the equation true.

In a one-step equation, only one operation is happening to the variable. That means the variable might be:

• added to a number, like \(x + 4 = 9\)
• subtracted by a number, like \(x - 3 = 7\)
• multiplied by a number, like \(5x = 20\)
• divided by a number, like \(\frac{x}{6} = 3\)

The goal is to isolate the variable. That means getting the variable by itself on one side of the equation.

A helpful way to think about equations is to imagine a balanced scale. If both sides are equal, the scale is balanced. To keep it balanced, whatever you do to one side, you must also do to the other side.

This is called doing the same operation to both sides.

To solve one-step equations, we use inverse operations. Inverse operations undo each other.

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

For example:

• To undo + 5, use - 5.
• To undo - 3, use + 3.
• To undo \(\times 4\), use \(\div 4\).
• To undo \(\div 6\), use \(\times 6\).

Steps for Solving a One-Step Equation

1. Look at what is happening to the variable.
2. Choose the inverse operation to undo it.
3. Do that operation to both sides of the equation.
4. Check your answer by putting it back into the original equation.

Worked Example 1: Addition

Solve \(x + 7 = 15\).

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

Check the answer:

Substitute \(8\) for \(x\).

This is true, so \(x = 8\).

Worked Example 2: Subtraction

Solve \(n - 6 = 10\).

The variable has 6 subtracted from it. To undo subtracting 6, add 6 to both sides.

Check the answer:

This is true, so \(n = 16\).

Worked Example 3: Multiplication

Solve \(4m = 28\).

The variable is being multiplied by 4. To undo multiplying by 4, divide both sides by 4.

Check the answer:

This is true, so \(m = 7\).

Worked Example 4: Division

Solve \(\frac{y}{5} = 9\).

The variable is being divided by 5. To undo dividing by 5, multiply both sides by 5.

Check the answer:

This is true, so \(y = 45\).

Using One-Step Equations in Word Problems

Sometimes equations come from real-life situations.

Example: Mia has 13 stickers. After getting some more, she has 20 stickers. How many stickers did she get?

Let \(s\) be the number of stickers she got. The equation is:

Subtract 13 from both sides:

Mia got 7 stickers.

Common Mistakes to Avoid

• Only changing one side: Always do the same operation to both sides.
• Using the wrong inverse operation: If the variable is multiplied by a number, divide. If the variable is divided by a number, multiply.
• Not checking your answer: Substitute your answer back into the original equation to make sure it works.

Quick Tips

• If you see \(x + a = b\), subtract \(a\).
• If you see \(x - a = b\), add \(a\).
• If you see \(ax = b\), divide by \(a\).
• If you see \(\frac{x}{a} = b\), multiply by \(a\).

Summary

To solve a one-step equation, get the variable alone by using the inverse operation. Remember to do the same thing to both sides so the equation stays balanced.

Then check your answer by substituting it back into the equation. With practice, solving one-step equations becomes quick and easy.

Solving Two-Step Equations

In math, an equation is a number sentence that uses an equals sign. It shows that two sides have the same value.

A variable is a letter that stands for an unknown number. For example, in the equation \(3x + 5 = 17\), the letter \(x\) is the variable.

A two-step equation is an equation that takes two steps to solve. We solve it by undoing the operations around the variable, one step at a time.

The big goal is to isolate the variable. That means getting the variable by itself on one side of the equation.

To do that, we use opposite operations:

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

Think of solving an equation like unwrapping a gift. You remove each layer in the reverse order that it was put on.

How to solve a two-step equation

1. Look at what is being done to the variable.
2. Undo the addition or subtraction first.
3. Then undo the multiplication or division.
4. Check your answer by substituting it back into the original equation.

Why do we usually undo addition or subtraction first? Because the number added or subtracted is usually the outer step. After removing it, the multiplication or division is easier to undo.

Important rule: Whatever you do to one side of the equation, you must do to the other side too. This keeps the equation balanced.

For example, if you subtract 4 from the left side, you must also subtract 4 from the right side.

Example 1: Addition, then multiplication

Solve:

Step 1: Undo the addition of 4 by subtracting 4 from both sides.

Step 2: Undo the multiplication by 3 by dividing both sides by 3.

Check the answer:

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

Example 2: Subtraction, then multiplication

Solve:

Step 1: Undo the subtraction of 7 by adding 7 to both sides.

Step 2: Undo the multiplication by 5 by dividing both sides by 5.

Check the answer:

So, the solution is \(y = 5\).

Example 3: Division, then addition

Solve:

Step 1: Undo the addition of 6 by subtracting 6 from both sides.

Step 2: Undo the division by 4 by multiplying both sides by 4.

Check the answer:

So, the solution is \(m = 12\).

Example 4: Division, then subtraction

Solve:

Step 1: Undo the subtraction of 2 by adding 2 to both sides.

Step 2: Undo the division by 3 by multiplying both sides by 3.

Check the answer:

So, the solution is \(n = 24\).

A pattern to remember

• If the equation looks like \(ax + b = c\), subtract or add \(b\) first, then divide by \(a\).
• If the equation looks like \(\frac{x}{a} + b = c\), subtract or add \(b\) first, then multiply by \(a\).

Common mistakes to avoid

• Doing steps in the wrong order: Undo addition or subtraction before multiplication or division.
• Forgetting to do the same thing to both sides: Keep the equation balanced.
• Skipping the check: Always put your answer back into the original equation.
• Mixing up opposite operations: To undo \(+ 8\), subtract 8. To undo \(\div 5\), multiply by 5.

How this connects to real situations

Two-step equations can help solve everyday problems. Suppose a club charges a \(\$3\) sign-up fee plus \(\$5\) each week. If the total cost is \(\$23\), you can write:

Subtract 3 from both sides:

Then divide by 5:

This means the cost was for 4 weeks.

Try these on your own

• \(2x + 7 = 15\)
• \(6p - 9 = 21\)
• \(\frac{q}{5} + 4 = 10\)
• \(\frac{r}{2} - 3 = 8\)

Answers

• \(x = 4\)
• \(p = 5\)
• \(q = 30\)
• \(r = 22\)

Summary

To solve a two-step equation, undo the operations around the variable in reverse order. Usually, that means undoing addition or subtraction first, then undoing multiplication or division.

Always do the same operation to both sides of the equation so it stays balanced. Then check your answer by putting it back into the original equation.

Writing Equations to Model Reality means turning a real-life situation into a math sentence with an unknown number. This helps us describe what is happening and find missing values.

An equation is a math sentence that shows two amounts are equal. It includes an unknown, usually written with a letter like \(x\), \(n\), or \(t\).

For example, if you have 3 apples and then get some more, and now you have 8 apples, you can write the equation \(3 + a = 8\). The letter \(a\) stands for the number of apples you got.

When we write equations from word problems, we are building a math model of the situation. A model is a way to represent something real using math.

Step 1: Understand the situation. Read carefully and ask: What is happening? What number is unknown? What numbers are given?

Step 2: Choose a variable. A variable is a letter that stands for the unknown amount. You might write, “Let \(x\) be the number of tickets,” or “Let \(m\) be the missing number of miles.”

Step 3: Look for clue words and relationships. Word problems often describe actions and comparisons. These can help you decide which operation to use.

• Addition: total, sum, in all, altogether, more than, increased by
• Subtraction: difference, left, fewer than, decreased by, remain
• Multiplication: groups of, times, each, product
• Division: shared equally, per, quotient, split into
• Equals: is, are, was, were, gives, results in

Step 4: Write the equation. Use the numbers, variable, and operations to match the story.

Step 5: Check whether the equation makes sense. Ask yourself: Does the equation match the words? Does the variable represent the correct unknown? Are both sides of the equation showing equal amounts?

There are two common kinds of equations you will write in 6th Grade:

• One-step equations, such as \(x + 7 = 19\) or \(4x = 28\)
• Two-step equations, such as \(3x + 5 = 20\) or \(x/4 - 2 = 6\)

Let’s look closely at how real situations become equations.

Part A: Writing one-step equations

In a one-step situation, the unknown is changed once by adding, subtracting, multiplying, or dividing.

If a problem says, “Mia had some stickers. She bought 6 more and now has 15,” the unknown is how many she had at first. Let \(s\) be the starting number of stickers. Since she bought 6 more, we add 6. Now she has 15, so the equation is \(s + 6 = 15\).

If a problem says, “A rope is cut into 5 equal pieces. Each piece is 3 feet long,” the unknown is the total length of the rope. Let \(r\) be the total length. Since 5 equal pieces of 3 feet each make the whole rope, the equation can be written as \(r \div 5 = 3\) or \(r/5 = 3\).

Part B: Writing two-step equations

In a two-step situation, the unknown is changed twice. You must pay attention to the order of the actions in the story.

For example, “A gym charges a \$12 sign-up fee and \$8 each class. The total cost is \$44.” The unknown is the number of classes. Let \(c\) be the number of classes. The cost has two parts: the sign-up fee and the class cost. The equation is:

$$8c + 12 = 44$$

This equation says: class cost plus sign-up fee equals total cost.

Be careful with words like “more than” and “less than.” These can be tricky.

“7 more than a number” means start with the number, then add 7: \(x + 7\).

“7 less than a number” means start with the number, then subtract 7: \(x - 7\).

“A number is 7 less than 20” means the number is \(20 - 7\), but if you are writing an equation with an unknown, you must match the sentence carefully.

Worked Example 1: One-step addition equation

Problem: Carlos had some marbles. He won 9 more marbles and now he has 21 marbles. Write an equation.

Step 1: Choose a variable. Let \(m\) be the number of marbles Carlos had at first.

Step 2: Show what happened. He won 9 more, so add 9.

Step 3: Write the total. Now he has 21.

The equation is:

$$m + 9 = 21$$

This equation models the real situation.

Worked Example 2: One-step multiplication equation

Problem: There are 6 bags of pencils. Each bag has the same number of pencils. Altogether, there are 42 pencils. Write an equation.

Step 1: Let \(p\) be the number of pencils in each bag.

Step 2: There are 6 equal groups, so multiply by 6.

Step 3: The total is 42.

The equation is:

$$6p = 42$$

This means 6 groups of \(p\) pencils make 42 pencils.

Worked Example 3: Two-step equation with addition

Problem: A movie theater charges \$5 for popcorn and \$3 for each drink. A family spends \$17 in all. Write an equation for the number of drinks.

Step 1: Let \(d\) be the number of drinks.

Step 2: The drinks cost \$3 each, so that part is \(3d\).

Step 3: Add the \$5 for popcorn.

Step 4: The total is \$17.

The equation is:

$$3d + 5 = 17$$

This equation matches the story: drink cost plus popcorn cost equals total cost.

Worked Example 4: Two-step equation with subtraction

Problem: Lena had some money. She spent \$4, then earned \$10, and now she has \$18. Write an equation.

Step 1: Let \(x\) be the amount of money Lena had at first.

Step 2: She spent \$4, so subtract 4: \(x - 4\).

Step 3: Then she earned \$10, so add 10: \(x - 4 + 10\).

Step 4: Now she has \$18.

The equation is:

$$x - 4 + 10 = 18$$

You could also simplify the left side to write:

$$x + 6 = 18$$

Both equations model the same situation.

How to tell whether to use the starting amount or the final amount as the variable

Sometimes the unknown is the beginning amount. Sometimes it is the ending amount. Read the question carefully.

• If the problem asks, “How many did she start with?” the variable should represent the starting amount.
• If the problem asks, “How many does she have now?” the variable should represent the final amount.
• If the problem asks, “How many in each group?” the variable should represent one group.
• If the problem asks, “How many groups?” the variable should represent the number of groups.

Example of choosing the right variable

“Jada has 24 stickers. She puts them equally onto 6 pages. How many stickers go on each page?”

Let \(s\) be the number of stickers on each page. Since 6 pages each have \(s\) stickers, the equation is:

$$6s = 24$$

If you picked the wrong meaning for the variable, your equation would not match the story correctly.

Common mistakes to avoid

• Using the wrong operation. Read for meaning, not just clue words.
• Forgetting what the variable stands for. Always define your variable first.
• Mixing up the order. In two-step situations, follow the actions in the order they happen.
• Leaving out part of the story. Make sure every important number is used correctly.
• Writing an expression instead of an equation. An equation must include an equals sign.

Try thinking with questions like these:

• What number do I not know?
• What is happening to that number?
• What is the total or result?
• Which operation matches the situation?
• Does my equation tell the same story as the words?

Quick practice ideas

Here are a few situations and the equations that match them:

• “A number increased by 8 is 23.” → \(x + 8 = 23\)
• “4 times a number is 36.” → \(4n = 36\)
• “A number minus 5 equals 12.” → \(m - 5 = 12\)
• “A club charges \$7 each month plus a \$15 joining fee for a total of \$50.” → \(7x + 15 = 50\)

Summary

Writing equations to model reality means turning a real-world story into a math sentence with a variable. First, decide what the unknown is. Then use the words in the problem to choose operations and write an equation that matches the situation.

One-step equations model situations with one change, and two-step equations model situations with two changes. If you define your variable, follow the story carefully, and check that both sides are equal, you can write equations that correctly represent real life.

Independent and Dependent Variables

In math, a variable is a letter or symbol that stands for a number that can change. When two variables are connected, one may affect the other.

This is where independent variables and dependent variables come in. Learning the difference helps you read tables, write equations, and understand real-life situations in math.

Independent variable: the variable you choose or change.

Dependent variable: the variable that changes because of the independent variable.

You can think of it like this:

• The independent variable is the cause.
• The dependent variable is the result.

Another helpful way to remember it is:

• Independent = stands on its own.
• Dependent = depends on something else.

For example, if you buy notebooks that cost \(\$3\) each, the total cost depends on how many notebooks you buy.

• Number of notebooks = independent variable
• Total cost = dependent variable

If you buy \(n\) notebooks, then the cost \(c\) is:

$$c = 3n$$

In this equation, \(n\) is independent because you decide how many notebooks to buy. The cost \(c\) depends on \(n\).

How to tell which variable is which

1. Read the situation carefully.
2. Ask, What am I choosing or changing? That is usually the independent variable.
3. Ask, What changes because of that choice? That is usually the dependent variable.

Look for clue words such as:

• "depends on"
• "for each"
• "total"
• "cost of"
• "number of"

These clues often help you see what causes the change and what reacts to it.

Independent and dependent variables in tables

Many times, these variables are shown in a table. The independent variable is usually listed first, because it is the input. The dependent variable is usually listed second, because it is the output.

Here is a table showing the cost of movie tickets at \(\$5\) each.

In this table:

• Number of tickets is the independent variable.
• Total cost is the dependent variable.

The total cost changes when the number of tickets changes.

Independent and dependent variables in equations

An equation shows how the variables are related. Often, the dependent variable is written by itself on one side.

Example:

$$y = 4x$$

In this equation:

• \(x\) is the independent variable.
• \(y\) is the dependent variable.

Why? Because the value of \(y\) depends on the value of \(x\).

If \(x = 2\), then:

$$y = 4(2) = 8$$

If \(x = 5\), then:

$$y = 4(5) = 20$$

Each time \(x\) changes, \(y\) changes too.

Worked Example 1: Simple real-life situation

A bakery sells muffins for \(\$2\) each. Let \(m\) be the number of muffins and \(c\) be the total cost.

Equation:

$$c = 2m$$

Step 1: What do you choose?

You choose how many muffins to buy, so \(m\) is the independent variable.

Step 2: What changes because of that?

The total cost changes based on the number of muffins, so \(c\) is the dependent variable.

Answer:

• Independent variable: \(m\)
• Dependent variable: \(c\)

Worked Example 2: Using a table

A student reads books during the summer. The table shows how many days the student reads and how many pages are read.

Step 1: Which value changes first?

The number of days is what we look at first. As the days increase, the pages read increase.

Step 2: Which value depends on the other?

The number of pages read depends on how many days the student reads.

Answer:

• Independent variable: days
• Dependent variable: pages read

Worked Example 3: Finding variables in an equation

A taxi charges \(\$6\) plus \(\$2\) for each mile. Let \(m\) be the number of miles and \(c\) be the total cost.

Equation:

$$c = 2m + 6$$

Step 1: What can change?

The number of miles can change, so \(m\) is the independent variable.

Step 2: What depends on that change?

The total cost depends on the number of miles, so \(c\) is the dependent variable.

If the taxi travels \(3\) miles, then:

$$c = 2(3) + 6 = 6 + 6 = 12$$

If the taxi travels \(7\) miles, then:

$$c = 2(7) + 6 = 14 + 6 = 20$$

As miles increase, cost increases.

Worked Example 4: A situation with a limit

You have \(\$20\) to spend on pencils that cost \(\$4\) each. Let \(p\) be the number of pencils and \(c\) be the total cost.

The cost rule is:

$$c = 4p$$

Because you only have \(\$20\), the total cost must be no more than \(20\):

$$c \le 20$$

So:

$$4p \le 20$$

Which variable is independent?

You choose how many pencils to buy, so \(p\) is the independent variable.

Which variable is dependent?

The total cost depends on the number of pencils, so \(c\) is the dependent variable.

What values of \(p\) work?

Divide both sides by \(4\):

$$p \le 5$$

This means you can buy at most \(5\) pencils.

This example shows that independent and dependent variables are also important when working with inequalities, not just equations.

Common mistakes to avoid

• Mistake 1: Picking the dependent variable first just because it is written first in a sentence. Always ask which one changes because of the other.
• Mistake 2: Thinking both variables are independent. In these situations, one usually affects the other.
• Mistake 3: Forgetting that cost, total, distance, and amount are often dependent on something else.

Quick check questions

1. A plant grows taller each week. Which is independent: weeks or plant height?
2. If \(t = 3n\), which variable depends on the other?
3. A store charges \(\$8\) per game. Is the total cost independent or dependent?

Answers

1. Weeks is the independent variable. Plant height is the dependent variable.
2. \(t\) depends on \(n\), so \(n\) is independent and \(t\) is dependent.
3. The total cost is dependent.

Summary

Independent and dependent variables help us understand how two changing amounts are connected.

• The independent variable is the one you choose or change.
• The dependent variable is the one that changes because of it.

In tables, the independent variable is usually the input, and the dependent variable is usually the output.

In equations like \(y = 5x\), \(x\) is usually independent and \(y\) is usually dependent.

Whenever you are unsure, ask yourself: What causes the change, and what reacts to the change?

Writing and Graphing Inequalities

Sometimes in math, an exact answer is not enough. We may need to show that a number can be greater than, less than, or possibly equal to another number. That is when we use inequalities.

An equation shows that two amounts are equal, like \(x = 5\). An inequality shows that two amounts are not always equal, but one amount may be bigger or smaller than the other, like \(x > 5\) or \(x \le 5\).

In this lesson, you will learn how to:

• write inequalities from words and real-life situations,
• understand what the inequality symbols mean,
• graph inequalities on a number line, and
• tell whether to use an open circle or a closed circle.

1. Inequality Symbols

There are four main inequality symbols you need to know:

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

Examples:

• \(x < 7\) means \(x\) is any number less than 7.
• \(x > 2\) means \(x\) is any number greater than 2.
• \(x \le 4\) means \(x\) can be 4 or any number less than 4.
• \(x \ge 9\) means \(x\) can be 9 or any number greater than 9.

A helpful way to remember the signs \(<\) and \(>\) is that the open side faces the bigger number.

2. Turning Words into Inequalities

Many inequality problems begin with words. Look for clue phrases that tell you which symbol to use.

Common phrases include:

• less than  \(<\)
• greater than  \(>\)
• at most  \(\le\)
• at least  \(\ge\)
• no more than  \(\le\)
• no fewer than  \(\ge\)

Here are some examples of word phrases and inequalities:

• "A number is greater than 8" becomes \(n > 8\).
• "A number is less than or equal to 12" becomes \(n \le 12\).
• "A number is at least 3" becomes \(n \ge 3\).
• "A number is no more than 15" becomes \(n \le 15\).

3. What Do Inequalities Mean?

An inequality usually has many solutions, not just one. For example, the inequality \(x > 4\) has lots of solutions.

Some solutions are:

• 5
• 6
• 10
• 100

But 4 is not a solution, because \(x > 4\) means the number must be bigger than 4, not equal to 4.

For \(x \ge 4\), the number 4 is a solution, because \(\ge\) means greater than or equal to.

4. Graphing Inequalities on a Number Line

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

When graphing inequalities, there are two important parts:

• the circle on the number line, and
• the shading or arrow that shows the direction of the solutions.

Open circle: Use an open circle when the endpoint is not included. This happens with \(<\) or \(>\).

Closed circle: Use a closed circle when the endpoint is included. This happens with \(\le\) or \(\ge\).

Shading left: Shade to the left for numbers that are less than a value.

Shading right: Shade to the right for numbers that are greater than a value.

Quick guide:

• \(x < 5\): open circle at 5, shade left
• \(x \le 5\): closed circle at 5, shade left
• \(x > 5\): open circle at 5, shade right
• \(x \ge 5\): closed circle at 5, shade right

5. Worked Examples

Example 1: Write an inequality from words

Write an inequality for: A number \(n\) is less than 9.

Step 1: Find the key words. The words are less than.

Step 2: Use the correct symbol, \(<\).

Answer:

$$n < 9$$

This means \(n\) can be any number smaller than 9.

Example 2: Graph a simple inequality

Graph \(x \ge 3\).

Step 1: The symbol is \(\ge\), which means greater than or equal to.

Step 2: Put a closed circle at 3, because 3 is included.

Step 3: Shade to the right, because the values are greater than 3.

Answer: Closed circle at 3, shading right.

Some numbers in the solution set are 3, 4, 5, and 10.

Example 3: Write an inequality from a real-life situation

A ride at a fair says you must be at least 48 inches tall to ride. Let \(h\) be the height in inches.

Step 1: The phrase at least means \(\ge\).

Step 2: Write the inequality.

$$h \ge 48$$

This means a person can be 48 inches tall or taller.

Graph: Put a closed circle at 48 and shade to the right.

Example 4: No more than

You can spend no more than \(\$20\) at the school store. Let \(m\) be the amount of money you spend.

Step 1: The phrase no more than means \(\le\).

Step 2: Write the inequality.

$$m \le 20$$

This means you can spend 20 dollars or less.

Graph: Put a closed circle at 20 and shade to the left.

6. How to Check a Solution

You can test whether a number is a solution by replacing the variable with that number.

Example: Is 6 a solution to \(x < 10\)?

Replace \(x\) with 6:

$$6 < 10$$

This is true, so 6 is a solution.

Example: Is 10 a solution to \(x < 10\)?

Replace \(x\) with 10:

$$10 < 10$$

This is false, so 10 is not a solution.

7. Common Mistakes to Avoid

• Mixing up open and closed circles
  Remember: open circle means not included; closed circle means included.
• Shading the wrong direction
  Less than goes left. Greater than goes right.
• Forgetting important words
  Words like or equal to, at least, and no more than matter a lot.
• Thinking an inequality has only one answer
  Most inequalities have many possible solutions.

8. Helpful Steps to Follow

1. Read the words carefully.
2. Find the clue phrase, such as less than or at least.
3. Choose the correct inequality symbol.
4. Write the inequality with a variable.
5. Graph it on a number line.
6. Use an open or closed circle correctly.
7. Shade left or right based on the symbol.

Summary

Inequalities compare numbers that may be greater than, less than, or equal to each other. The symbols \(<\), \(>\), \(\le\), and \(\ge\) help us write these comparisons.

When graphing, use an open circle for \(<\) or \(>\), and a closed circle for \(\le\) or \(\ge\). Shade to the left for less than and to the right for greater than.

With practice, you can read a situation, write an inequality, and graph all of its possible solutions.

Solving One-Step Inequalities helps us find all the numbers that make a math statement true.

An inequality compares two amounts that are not always equal. Instead of only using the equals sign, inequalities use symbols like:

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

For example, in the inequality \(x + 3 < 10\), we are looking for every value of \(x\) that makes the statement true.

Unlike an equation, which often has one answer, an inequality usually has many possible answers.

What does it mean to solve an inequality?

To solve an inequality means to find the values of the variable that make the inequality true.

We solve one-step inequalities in almost the same way we solve one-step equations. We use inverse operations to undo what is happening to the variable.

Inverse operations are opposite operations:

• Addition and subtraction
• Multiplication and division

Main idea: Whatever you do to one side of an inequality, you must also do to the other side to keep it balanced.

For example, if you subtract 4 from the left side, you must also subtract 4 from the right side.

Important rule: If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

This is a very important inequality rule.

For example, if

and we divide both sides by \(-2\), the sign must flip:

Why does the sign flip?

Negative numbers change the order on the number line. For example, \(5 > 2\), but if we multiply both numbers by \(-1\), we get \(-5\) and \(-2\). Now \(-5 < -2\). The direction changes.

Steps for solving one-step inequalities

1. Find what operation is being done to the variable.
2. Use the inverse operation to undo it.
3. Do the same thing to both sides.
4. If you multiply or divide by a negative number, reverse the inequality sign.
5. Check your answer if needed.

Worked Example 1: Addition

Solve:

The variable has 6 added to it. Undo that by subtracting 6 from both sides.

So the solution is:

This means any number less than 8 will make the inequality true.

Some possible answers are \(7\), \(0\), and \(-3\).

Worked Example 2: Subtraction

Solve:

The variable has 4 subtracted. Undo that by adding 4 to both sides.

So the solution is:

This means 13 and any number greater than 13 will work.

Worked Example 3: Multiplication by a positive number

Solve:

The variable is multiplied by 3. Undo that by dividing both sides by 3.

The sign stays the same because we divided by a positive number.

Worked Example 4: Division by a negative number

Solve:

The variable is divided by \(-5\). Undo that by multiplying both sides by \(-5\).

Because we are multiplying by a negative number, we must reverse the inequality sign.

So the solution is:

Checking the answer

It is a good idea to test a value that should work.

For the inequality \(n < -15\), try \(n = -20\):

This is true, so the solution makes sense.

What if we forget to flip the sign?

Look again at:

If someone divides by \(-2\) but forgets to flip the sign, they might say:

But test \(x = 0\), which is greater than \(-4\):

That is false. So the sign must be reversed.

Quick guide by operation

• If the variable has a number added, subtract that number.
• If the variable has a number subtracted, add that number.
• If the variable is multiplied by a positive number, divide by that number.
• If the variable is divided by a positive number, multiply by that number.
• If you multiply or divide by a negative number, flip the inequality sign.

Common mistakes to avoid

• Treating an inequality exactly like an equation and forgetting that it can have many answers.
• Forgetting to do the same operation to both sides.
• Forgetting to reverse the sign when multiplying or dividing by a negative number.
• Mixing up the inequality symbols.

Number line idea

Inequalities can also be shown on a number line.

• \(x < 8\): all numbers to the left of 8
• \(y \ge 13\): 13 and all numbers to the right

If the solution includes the number, like \(\ge\) or \(\le\), use a closed dot.

If the solution does not include the number, like \(<\) or \(>\), use an open dot.

Practice thinking

Ask yourself these questions when solving:

• What is happening to the variable?
• What inverse operation will undo it?
• Am I multiplying or dividing by a negative number?
• Does the inequality sign need to flip?

Summary

To solve a one-step inequality, use inverse operations to isolate the variable.

Do the same operation to both sides to keep the inequality balanced.

If you multiply or divide by a negative number, always reverse the inequality sign.

Once solved, your answer represents all the values that make the inequality true.

Validating Solutions using Substitution means checking whether an answer really works.

Sometimes in math, we see a letter like \(x\) or \(n\). The letter stands for a number we do not know yet. After we find a possible answer, we should check it.

We can check by using substitution. Substitution means putting the number in place of the letter. Then we solve the math sentence and see if it is true.

This is an important habit because it helps us catch mistakes and feel sure that our answer is correct.

What does it mean for a solution to be valid?

A solution is an answer for the unknown number. A solution is valid if, after we substitute it into the equation, both sides are equal.

For example, in the equation \(x + 4 = 9\), the number \(5\) is a valid solution because:

$$5 + 4 = 9$$

Since both sides match, the answer works.

How to check an answer using substitution

1. Look at the equation.
2. Take the value you think is the answer.
3. Replace the letter with that number.
4. Solve the number sentence.
5. Check whether both sides are the same.

If both sides are the same, the solution is valid. If they are not the same, the solution is not valid.

Example 1: One-step addition equation

Check whether \(x = 6\) is a valid solution for:

$$x + 3 = 9$$

Step 1: Substitute \(6\) for \(x\).

$$6 + 3 = 9$$

Step 2: Solve the left side.

$$9 = 9$$

Step 3: Compare both sides.

Both sides are \(9\), so they are equal.

Answer: Yes, \(x = 6\) is a valid solution.

Example 2: One-step subtraction equation

Check whether \(n = 8\) is a valid solution for:

$$n - 5 = 2$$

Step 1: Substitute \(8\) for \(n\).

$$8 - 5 = 2$$

Step 2: Solve the left side.

$$3 = 2$$

Step 3: Compare both sides.

The sides are not equal.

Answer: No, \(n = 8\) is not a valid solution.

Let us try the correct value. If \(n = 7\), then:

$$7 - 5 = 2$$

$$2 = 2$$

So \(n = 7\) is valid.

Example 3: Multiplication equation

Check whether \(m = 4\) is a valid solution for:

$$3m = 12$$

In 4th Grade math, \(3m\) means \(3 \times m\).

Step 1: Substitute \(4\) for \(m\).

$$3 \times 4 = 12$$

Step 2: Solve.

$$12 = 12$$

Step 3: Compare both sides.

Both sides are equal.

Answer: Yes, \(m = 4\) is a valid solution.

What if someone said \(m = 5\)? Let us check:

$$3 \times 5 = 12$$

$$15 = 12$$

That is not true, so \(m = 5\) is not valid.

Example 4: Two-step equation

Check whether \(y = 3\) is a valid solution for:

$$2y + 1 = 7$$

Step 1: Substitute \(3\) for \(y\).

$$2 \times 3 + 1 = 7$$

Step 2: Solve in order.

First multiply:

$$6 + 1 = 7$$

Then add:

$$7 = 7$$

Step 3: Compare both sides.

Both sides are equal.

Answer: Yes, \(y = 3\) is a valid solution.

Now check \(y = 2\) for the same equation:

$$2 \times 2 + 1 = 7$$

$$4 + 1 = 7$$

$$5 = 7$$

This is false, so \(y = 2\) is not valid.

Tips for success

• Replace the letter carefully. Make sure you put the number everywhere the letter appears.
• Solve step by step. Do not rush.
• Check both sides. Ask: “Do they match?”
• If they do not match, the answer is not correct.

Why substitution is helpful

Substitution helps us test our thinking. Even if we think we solved the equation correctly, checking with substitution gives us proof.

It is like trying a key in a lock. If the key opens the lock, it is the right key. If it does not, we need another key. In math, if the substituted number makes the equation true, it is the right answer.

Practice thinking

Here are some equations you can check on your own:

• Is \(x = 2\) valid for \(x + 5 = 7\)?
• Is \(a = 9\) valid for \(a - 4 = 5\)?
• Is \(b = 6\) valid for \(2b = 12\)?
• Is \(c = 4\) valid for \(c + 3 = 10\)?

Brief Summary

To validate a solution using substitution, replace the letter with the number, solve the equation, and see whether both sides are equal. If both sides match, the solution is valid. If they do not match, the solution is not valid. Checking your answer this way helps you be sure your math is correct.