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

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

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

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.

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

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

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

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

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

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

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 + 3 < 2x – 5 –2x –2x a 4x + 3 < –5 a – 3 –3 a 4x < –8 a 4 4 a x < –2 a -1 1 2 3 4 -6 -5 -4 -3 -2

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

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 + 7  19 7 7 4x  12 4 4 x  3  The sign changed direction because both sides were divided by a negative number.

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

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?

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