1. 8/8/20151 6.5 Inequalities

2. 8/8/2015 2 Bumper Cars You must be at least 130cm tall to ride the bumper cars. This can be represented by the inequality h  130. In the Real World Solving Inequalities 6. 5

3. 8/8/2015 3 The solution of an inequality is the set of numbers that you can substitute for the variable to make the inequality true. An inequality is a mathematical sentence formed by placing an inequality symbol between two expressions. Bumper Cars You must be at least 130cm tall to ride the bumper cars. This can be represented by the inequality h  130. In the Real World The graph of an inequality in one variable is the set of points on a number line that represents the solution of the inequality. Solving Inequalities 6 5.

4. 8/8/2015 4 Graphing Inequalities EXAMPLE 1 x < 2All numbers less than 2 1–3–2–1032–44 Solving Inequalities 6 5.

5. 8/8/2015 5 Graphing Inequalities EXAMPLE 1 x < 2 All numbers less than 2 x  –1 All numbers less than or equal to –1 1–3–2–1032–441–3–2–1032–44 Solving Inequalities 6 5.

6. 8/8/2015 6 Graphing Inequalities EXAMPLE 1 x < 2 All numbers less than 2 x  –1 All numbers less than or equal to –1 x > 0 All numbers greater than 0 1–3–2–1032–441–3–2–1032–441–3–2–1032–44 Solving Inequalities 6 5.

7. 8/8/2015 7 Graphing Inequalities EXAMPLE 1 x < 2 All numbers less than 2 x  –1 All numbers less than or equal to –1 x > 0 All numbers greater than 0 x  –2 All numbers greater than or equal to –2 1–3–2–1032–441–3–2–1032–441–3–2–1032–441–3–2–1032–44 Solving Inequalities 7 6.

8. 8/8/2015 8 Solving Inequalities Equivalent inequalities are inequalities that have the same solution. You can produce an equivalent inequality in the following ways: Add or subtract the same number on each side. Multiply or divide each side by the same positive number. 1 < 3 2 < 6 Multiply each side by 2. 6 5.

9. 8/8/2015 9 Solving Inequalities Solving an Inequality EXAMPLE 2 Solve d – 2  1. d – 2  1 Add 2 to each side. Write inequality. Simplify. d – 2 + 2  1 + 2 d  3 To graph d  3, use a closed dot and draw the arrow pointing to the left. 2–2–10143–35 Then graph the solution. 6 5.

10. 8/8/2015 10 Solving Inequalities Solving an Inequality EXAMPLE 2 Solve d – 2  1. d – 2  1 Add 2 to each side. Write original inequality. Simplify. Write original equation. Substitute 0 for d. Solution checks. d – 2 + 2  1 + 2 d  3 To graph d  3, use a closed dot and draw the arrow pointing to the left. C HECK To check the solution d  3, choose any number less than or equal to 3 to substitute for d. The check below uses d = 0. d – 2  1 –2  1 2–2–10143–35 Then graph the solution. 0 – 2  1 ? 6 5.

11. 8/8/2015 11 Solving Inequalities Solving an Inequality EXAMPLE 3 Solve 5 + y <13 5 + y < 13 Subtract 5 to each side. Write original inequality. Simplify. 5 + y - 5 < 13 - 5 y < 8 To graph y < 8, use an open dot and draw the arrow pointing to the left. 7 345698210 Then graph the solution. 6 5.

12. 8/8/2015 12 Solving Inequalities Solving an Inequality EXAMPLE 4 Solve a – 3 > -5 a – 3 > -5 Add 3 to each side.. Write original inequality. Simplify. a – 3 + 3 > -5 + 3 a > -2 To graph a > -2 use a closed dot and draw the arrow pointing to the right. Then graph the solution. 6 5. 2–2–10143–35

13. 8/8/2015 13 Solving Inequalities Solving an Inequality EXAMPLE 5 Solve 3 t > 12 3 t > 12 Divide each side by 3.. Write original inequality. Simplify. 3 t > 12 t > 4 To graph t > 4 use an open dot and draw the arrow pointing to the right. Then graph the solution. 6 5. 8 4 567109311 3

14. 8/8/2015 14 Solving Inequalities Solving an Inequality EXAMPLE 6 Solve 3x + 5 < 23 3x + 5 < 23 Subtract 5 on each side Write original inequality. Simplify. 3x + 5 < 23 3x < 18 To graph x < 6 use an open dot and draw the arrow pointing to the left. Then graph the solution. 6 5. 8 4 567109311 -5 x < 6 Divide both sides by 3 and simplify Assignment: p. 248 9 - 22