Solutions to Equations and

InequalitiesLesson 7.01

After completing this lesson, you will be able to say:

• I can use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Key Terms

• Equation: mathematical sentence that shows two expressions are equal using the equal sign

• Solution: Any value substituted for a variable that makes the mathematical sentence true

Example of an equation and solution

Balancing an equation

• You can find solutions to an equation by using a balance scale.

• When an equation is balanced the scales are equal on both sides

Balancing the Scale

The balance scale is not balanced, what can you do to balance the scale?

If we remove one block from the left side, the scale will be balanced

An equation is two expressions that are equal to each other. Just as you balanced the scales, you were proving that the left side was equal to the right side. Therefore, you created a true statement. For example, 4 = 4 is a true statement, whereas, 5 = 4 is a false statement.

Balancing the scales

You can determine if an equation is true or false by substituting a value in for the variable. • When the left side equals the right side, the equation is balanced. This

means the equation is true. • When the left side and right side are not equal, the equation is unbalanced.

This means the equation is false.

Balancing the scales

3 + 4 = 7 is a true statement.

7 = 7

4 + 4 = 7 is a false statement.

8 ≠ 7

Therefore, 3 is the only solution that makes the statement true.

Try it

Is 4 a solution to the equation 5x = 20?

Check your work

Check by substituting 4 for the variable and simplifying.

5(4) = 20 Substitute the variable with the given value and simplify.

20 = 20 Is this a true statement?

Yes! Therefore, 4 is a solution of the equation 5x = 20, because it makes a true statement.

InequalitiesInequality:

A mathematical sentence that shows a comparison between two expressions using the less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) symbols.

Inequalities

The inequality symbols “is less than or equal to” (≤) and “is greater than or equal to” (≥) are like two symbols in one.

Think of a statement that uses one of these symbols as a combination of an inequality and an equation. If either the inequality or the equation is true, then the entire statement is true.

Inequalities - Examples

Is x ≤ 8 true, when x equals 5?

The statement 5 ≤ 8 is true if either the statement 5 = 8 or the statement 5 < 8 is true.

5 = 8 is false.

5 < 8 is true.

Because 5 < 8 is true, the inequality 5 ≤ 8 is true because 5 is less than 8.

The solution can be less than or equal to as it cannot be both.

Inequalities - Example

Try It!

Barnabas believes that x = 7 is a solution to the inequality 4x + 5 > 34. Is he correct?

Check your work

4x + 5 > 344(7) + 5 > 34 Substitute 7 into the variable

of the inequality.28 + 5 > 34 Simplify.33 > 34 Is this a true statement?

This is not a true statement because 33 is not greater than 34. Therefore, 7 is not a solution of the inequality.

Try It

Why is x = 4 not a valid solution to the inequality 7x + 5 > 33?

Check your work

When you substitute x = 4 into the inequality and simplify, the statement is not true.

7x + 57(4) + 5 > 33 Substitute and simplify.33 > 33 Is this a true statement?

Because 33 is not greater than 33, x = 4 is not a solution of the inequality.

Sets of Numbers

Because inequalities compare two expressions, there are multiple values that can make the statement true. Sometimes, you may have to check multiple values that are presented in a set.

Sets of Numbers - Example

Which value or values from the set {1, 3, 5} make the inequality 4x + 8 > 12 a true statement? How do you know?

Substitute each value from the set into the inequality to see which values make a true statement.

Substitute 1 into the inequality and simplify4x + 8 > 12

4(1) + 8 > 12

4 + 8 > 12

12 > 12

Is this a true statement?This is not a true statement. The value 12 is not greater than 12, so x = 1 is not a solution.

Substitute 3 into the inequality and simplify4x + 8 > 12

4(3) + 8 > 12

12 + 8 > 12

20 > 12

Is this a true statement?This is a true statement. The value 20 is greater than 12, so x = 3 is a solution.

Substitute 5 into the inequality and simplify4x + 8 > 12

4(5) + 8 > 12

20 + 8 > 12

28 > 12

Is this a true statement?This is a true statement. The value 28 is greater than 12, so x = 5 is a solution.

Try it

Check your work

Substitute 25 into the inequality and simplify

Is this a true statement?This is not a true statement. so x = 25 is not a solution.

Substitute 45 into the inequality and simplify

Is this a true statement?This is a true statement. so x = 45 is a solution.

Substitute 55 into the inequality and simplify

Is this a true statement?This is a true statement. so x = 55 is a solution.

Try it!

Erica is creating a rectangular garden with an area less than or equal to 100 square feet. Erica can use the inequality LW ≤ 100 represent the area of the garden, where L is the length and W is the width. If the length of the garden has to be 25 feet, can she make the garden 5 feet wide?

Check your work

Substitute L = 25 and W = 5 into the inequality.

25(5) ≤ 100 Substitute and simplify.

125 ≤ 100 Is this a true statement?

This is not a true statement. Because 125 is not equal to or less than 100, Erica cannot make the garden 5 feet wide.

Now that you completed this lesson, you should be able to say:

• I can use substitution to determine whether a given number in a specified set makes an equation or inequality true.