1. Core Focus on Rational Numbers & Equations
Lesson 1.8
Core Focus on Rational Numbers & Equations
Linear Inequalities in One Variable

2. Warm-Up Solve each equation for the variable. 1. h = 4 2. 3. y = 2
x = 28

3. Solve inequalities with one variable.
Lesson 1.8
Linear Inequalities
Solve inequalities with one variable.

4. Jackie has run at most 200 miles.
Vocabulary
Inequality
A mathematical sentence that contains < , >,  , or  to show a relationship between quantities.
Nathan has more than
$10 in his wallet.
n > $10.00
Jackie has run at most 200 miles.
j ≤ 200
Good to Know!
Inequalities have multiple answers that can make the statement true. In Nathan’s example, he might have $20 or $100, all that is known for certain is that he has more than $10 in his wallet. There are an infinite number of possibilities that make the statement n > $10.00 true.

5. Inequality Symbols
> “greater than” < “less than”  “greater than or equal to”  “less than or equal to”

6. Example 1
Write an inequality for each statement. a. Carla’s weight (w) is greater than 100 pounds. The key words are “greater than.” w > 100 b. Vicky has at most $500 in her savings account. Let m represent the amount of money in Vicky’s account. The key words are “at most.” This means m ≤ $500 she has less than or equal to $500. c. Quinton’s age is greater than 40 years old. Let a represent Quinton’s age. The key words are “greater than.” a > 40

7. Extra Example 1
Write an inequality for each statement. a. Sharon earns at least 8 dollars (d) per baby-sitting job. b. Kenny does less than 10 hours (h) of homework per week. c. Rayanna is more than 48 inches (i) tall.
d ≥ 8
h < 10
i > 48

8. Graphing Inequalities
Solutions to an inequality can be graphed on a number line. For < or >, use an OPEN circle to graph the inequality: x > 1 x < 1 For  or , use a CLOSED circle to graph the inequality: x  1 x  1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1

9. Example 2
Solve the inequality and graph its solution on a number line. Subtract 2 from both sides of the inequality. Multiply both sides of the inequality by 4. Graph the solution on a number line. Use a closed circle.
Inequalities are solved using properties similar to those you used to solve equations.
– 2
– 2
4 
 4 4 5 6 7 8 9 -1 1 2 3

10. Extra Example 2
Solve the inequality and graph its solution on a number line. 4x + 5 > 21
x > 4

11. Example 3
Solve the inequality and graph its solution on a number line.
6x + 3 < 2x – 5
Subtract 2x from each side of the inequality.
Subtract 3 from each side.
Divide both sides by 4.
Graph the solution on a number line.
Use an open circle.
6x < 2x – 5
–2x –2x a
4x + 3 < – a
– – a
4x < – a a
x < – a -1 1 2 3 4 -6 -5 -4 -3 -2

12. Extra Example 3
Solve the inequality and graph its solution on a number line. 2x − 5 ≤ 4x − 7
x ≥ 1

13. Example 4
Solve the inequality.  4x + 7  19 Subtract 7 from each side of the inequality. Divide both sides by  4. Since both sides were divided by a negative, flip the inequality symbol.
4x  19
 7
4x  12
 4
x  3 
The sign changed direction because both sides were divided by a negative number.

14. Extra Example 4
Solve the inequality 9 ≥ −3x + 15.
x ≥ 2

15. Communication Prompt
Number lines are used to give a visual picture of an inequality statement. What is another situation in math where a visual is used to show math?

16. Exit Problems 1. Write an inequality for the graph. x ≥ –3
2. Write an inequality for the statement, “Lance walked more than 2 miles (m).”
3. Solve the inequality and graph the solution: 2x + 7 < 3.
x ≥ –3 -1 1 2 -6 -5 -4 -3 -2
m > 2 -1 1 2 -6 -5 -4 -3 -2
x < –2