Do Now (do this on the notebook paper, please) Graph each inequality. 1. p ≤ 2 2. n < 3 –3 –2 –1 0 1 2 3 3. a ≥ –4 –6 –4 –2 0 2 4 6 4. x > 0 –6 –4 –2 0 2 4 6

Symbol Meaning Word Phrases An inequality states that two quantities either are not equal or may not be equal. An inequality uses one of the following symbols: Symbol Meaning Word Phrases < > ≤ ≥ is less than Fewer than, below is greater than More than, above is less than or equal to At most, no more than is greater than or equal to At least, no less than

An inequality that contains a variable is an algebraic inequality An inequality that contains a variable is an algebraic inequality. A value of the variable that makes the inequality true is a solution of the inequality. An inequality may have more than one solution. Together, all of the solutions are called the solution set. You can graph the solutions of an inequality on a number line. If the variable is “greater than” or “less than” a number, then that number is indicated with an open circle.

a > 5 b ≤ 3 This open circle shows that 5 is not a solution. If the variable is “greater than or equal to” or “less than or equal to” a number, that number is indicated with a closed circle. This closed circle shows that 3 is a solution. b ≤ 3

Course Inequalities -3 < b is the same as b > -3. Reading Math

You can find solution sets of inequalities the same way you find solutions of equations, by isolating the variable. Solve. Then graph the solution set on a number line. n – 7 ≤ 15 n – 7 ≤ 15 + 7 + 7 n ≤ 22 -44 -22 22 44 66 –88 -66

Solve. Then graph the solution set on a number line. - 4 -4 b ≥ 6 –4 –2 0 2 4 6 8 10

Solve. Then graph the solution set on a number line. 3 3 b < 5 –4 –2 0 2 4 6 8 10

Solve. Then graph the solution set on a number line.     y > 1 –4 –2 0 2 4 6 8 10

When you multiply or divide both sides of an inequality by the same positive number, the statement will still be true. However, when you multiply or divide both sides of an inequality by the same negative number, you need to reverse the direction of the inequality symbol for the statement to be true!

–4 < 2 (3)(–4) < (3)(2) –12 < 6 –4 < 2 (–3)(–4) > (–3)(2) 12 > –6

c 4 ≤ –4 c 4 ≤ –4 c 4 (4) ≤ (–4) (4) Multiply both sides by 4. c ≤ –16 Solve. c 4 ≤ –4 c 4 ≤ –4 c 4 (4) ≤ (–4) (4) Multiply both sides by 4. c ≤ –16

Multiply both sides by –4 and reverse the inequality symbol. –4 Solve. t > 0.3 –4 t –4 > 0.3 t (–4) < (–4)0.3 Multiply both sides by –4 and reverse the inequality symbol. –4 t < –1.2

n 6 ≤ –5 n 6 ≤ –5 n 6 (6) ≤ (–5) (6) Multiply both sides by 6. n ≤ –30 Solve. n 6 ≤ –5 n 6 ≤ –5 n 6 (6) ≤ (–5) (6) Multiply both sides by 6. n ≤ –30

Solve. Check your answer. Divide both sides by 5. 5 5 23 5 3 5 , or 4 a ≥ Check 5a ≥ 23 3 5 ? 5 is greater than 4 . 5(5) 23 ≥ Substitute 5 for a. ? 25 23 ≥

Additional Example 2B: Solving Inequalities by Dividing Course 2 Solving Inequalities by Multiplying or Dividing Additional Example 2B: Solving Inequalities by Dividing Solve. Check your answer. 192< -24b 192 < -24b Divide both sides by –24, and reverse the inequality symbol. –24 –24 -8 > b Check 192 < -24b 192< -24(-10) ? -10 is less than –8. Substitute -10 for b. ? 192 < 240

Solving Inequalities by Multiplying or Dividing Course 2 Solving Inequalities by Multiplying or Dividing Check It Out: Example 2B Solve. Check your answer. 85 < -17b 85 < -17b Divide both sides by –17, and reverse the inequality symbol. –17 –17 -5 > b Check 85 < -17b 85 < -17(-6) ? -6 is less than –5. Substitute -6 for b. ? 85 < 102

Additional Example 3: Application It cost Josh $85 to make candles for the craft fair. How many candles must he sell at $4.00 each to make a profit? Since profit is the amount earned minus the amount spent, Josh needs to earn more than $85. Let c represent the number of candles that must be sold. 4c > 85 Write an inequality. 4c > 85 Divide both sides by 4. 4 4 c > 21.25 Josh cannot sell 0.25 candle, so he needs to sell at least 22 candles, or more than 21 candles, to earn a profit.

Check It Out: Example 3 It cost the class $15 to make cookies for the bake sale. How many cookies must they sell at 10¢ each to make a profit? Since profit is the amount earned minus the amount spent, the class needs to earn more than $15. Let c represent the number of cookies that must be sold. 0.10c > 15 Write an inequality. 0.10c > 15 Divide both sides by .10. 0.10 0.10 c > 150 The class must sell more than 150 cookies to make a profit.

Insert Lesson Title Here Lesson Quiz Solve. 1. 2. Solve. Check each answer. 3. 18w < 4 4. –4f > 36 5. It cost a candle company $51 to make candles. How many candles must it sell at $7 apiece to make a profit? 6. -2x – 2 < 10 s 9 > 12 s > 108 b –14 > 6 b < –84 w < 2 9 f < –9 least 8 candles more than 7 candles, or at