Solving One-Step Inequalities Algebra 1 ~ Chapter 6.1 and 6.2 Solving One-Step Inequalities

Recall that statements with greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) are inequalities. Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations.

The solution set is {all numbers Ex. 1 - Solve the inequality and graph the solutions. x + 12 < 20 –12 –12 The solution set is {all numbers less than 8}. x < 8 The circle at 8 is open. This shows that 8 is NOT included in the inequality. –10 –8 –6 –4 –2 2 4 6 8 10 The heavy arrow pointing to the left shows that the inequality includes all #s less 8. Check your solution??

Ex. 2 - Solve the inequality and graph the solutions. +5 +5 d > –2 Example check: d = 0 d – 5 > -7 0 – 5 > -7 -5 > -7 TRUE! Example check: d = -6 d – 5 > -7 -6 - 5 > -7 -11 > -7 FALSE! –10 –8 –6 –4 –2 2 4 6 8 10

Ex. 3 – Solving an Inequality with Variables on both sides Solve 12x – 4 ≤ 13x -12x -12x -4 ≤ 1x x ≥ -4

x > –6 Ex. 4 - Solve the inequality and graph the solutions. CHECK 7(0) > -42 0 > -42 TRUE! 7x > –42 CHECK 7x > -42 7(-10)>-42 -70 > -42 FALSE! x > –6 –10 –8 –6 –4 –2 2 4 6 8 10

Ex. 5 - Solve the inequality and graph the solutions. Check: 3(2) ≤ 3 6 ≤ m (or m ≥ 6) 2 4 6 8 10 12 14 16 18 20

Ex. 6 - Solve the inequality and graph the solutions. Since r is multiplied by , multiply both sides by the reciprocal of . r < 16 2 4 6 8 10 12 14 16 18 20

What happens when you multiply or divide both sides of an inequality by a negative number? Look at the number line below. –b –a a b a < b –a –b You can tell from the number line that –a > –b. Multiply both sides by –1. b > –a –b a Multiply both sides by –1. You can tell from the number line that –b < a. > < Notice that when you multiply (or divide) both sides of an inequality by a negative number, you must REVERSE the inequality symbol.

A real number example… 3 < 8 -1(3) ? -1(8) -3 -8 > Start off with the # 8 is greater than the # 3. TRUE! 3 < 8 -1(3) ? -1(8) -3 -8 If I multiply both sides by a negative # (-1 in this case)… In order to keep this inequality TRUE, what must I do to the inequality symbol? > REVERSE or FLIP the inequality sign!!

Caution! Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

Ex. 7 - Solve the inequality and graph the solutions. Since x is multiplied by –12, divide both sides by –12. Change > to <. x < –7 CHECK -12x > 84 -12(-12) > 84 144 > 84 TRUE! –10 –8 –6 –4 –2 2 4 6 –12 –14 –7

Ex. 8 - Solve the inequality and graph the solutions. Since x is divided by –3, multiply both sides by –3. Change to . 24  x (or x  24) 16 18 20 22 24 10 14 26 28 30 12

Multiply both sides by –1 to make x positive. Change  to . Ex. 9 - Solve the inequality and graph the solutions. Check your answer. 10 ≥ –x Multiply both sides by –1 to make x positive. Change  to . –1(10) ≤ –1(–x) CHECK 10 ≥ –x 10 ≥ -(4) 10 ≥ -4 TRUE! –10 ≤ x (or even better x ≥ -10) –10 –8 –6 –4 –2 2 4 6 8 10

Ex. 10 – Define a variable and write an inequality for each problem Ex. 10 – Define a variable and write an inequality for each problem. You do not need to solve the inequality. a.) A number decreased by 8 is at most 14. b.) A number plus 7 is greater than 2. c.) Half of a number is at least 26. n – 8 ≤ 14 n + 7 > 2 ½n ≥ 26

Lesson Review Solve each inequality and graph the solutions. 1. 13 < x + 7 x > 6 2. –6 + h ≥ 15 h ≥ 21 3. 4. –5x ≥ 30 x > 20 x ≤ –6

Assignment Worksheet 6-1 & 6-2 (Front and Back) Pages 321-322 #’s 14-32 (evens), 40-46 (evens) Pages 329-330 #’s 14-36 (evens), 40-46 (evens)