2 6 Solving Linear Inequalities Inequality x 2

2. 6 Solving Linear Inequalities

Inequality x > 2 Graph | | | ( | | -1 0 1 2 3 4 x < 2 x ≥ 2 x ≤ 2 Interval Notation (2, ) | | | ) | | -1 0 1 2 3 4 (- , 2) | | | [ | | -1 0 1 2 3 4 [2, ) | | | ] | | -1 0 1 2 3 4 (- , 2]

Note: • If the inequality symbol is > or < use ( or ) – the solution number is not included in the solution set • If the inequality symbol is ≥ or ≤ use [ or ] – the solution number IS included in the solution set

Addition Property of Inequalities Adding or subtracting the same quantity from each side of an inequality will not change the solution set. EX: 3 < 9 3 + 1 < 9 + 1 3 – 1 < 9 – 1 4 < 10 2 < 8 True

Ex: Solve and graph the solution set: x + 5 > 12 x + 5 – 5 > 12 – 5 x > 7 | | | ( | | 4 5 6 7 8 9 Solution: (7, )

Multiplication Property of Inequalities v Multiplying/Dividing both sides of an inequality by a POSITIVE number does NOT change the solution set v Mult/Div both sides of an inequality by a NEGATIVE number REQUIRES the inequality symbol to be REVERSED to produce an equivalent inequality REVERSE SYMBOL EX: 3 < 9 3 ۰ 1 < 9 ۰ 1 3 ۰ (– 1) < 9۰ (– 1) 3 ۰ (– 1) > 9۰ (– 1) 3 < 9 – 3 < – 9 – 3 > – 9 True False True

Ex: Solve and graph the solution set. 7 y < 14 7 7 y < 2 | | ) | | | 0 1 2 3 4 5 Solution: (- , 2)

Ex: Solve and graph the solution set. -3 a < 12 -3 a > 12 -3 -3 divide by neg. # so reverse ineq. sym a > -4 | ( | | -5 -4 -3 -2 -1 0 Solution: (-4, )

Ex: Solve and graph the solution set. -5 x + 6 ≤ -9 -5 x + 6 – 6 ≤ -9 – 6 -5 x ≤ -15 -5 x ≥ -15 -5 divide by neg. # so reverse ineq. sym x ≥ 3 | | | [ | | 0 1 2 3 4 5 Solution: [3, )

Unusual Stuff Ex: Solve. 3 x – 5 < 3(x – 2) 3 x – 5 < 3 x – 6 3 x – 5 – 3 x < 3 x – 6 – 3 x -5 < -6 No variables remain and stmt. is FALSE, so no soln. Answer: ø

Ex: Solve. 5(x + 4) > 5 x + 10 5 x + 20 > 5 x + 10 5 x + 20 – 5 x > 5 x + 10 – 5 x 20 > 10 No variables remain and stmt. is TRUE, so all real #s are solns. Answer: (- , )

Application Problems WORDS SYMBOLS x is at least 12 x ≥ 12 x is more than 12 x > 12 x is at most 13 x ≤ 13 x is less than 13 x < 13

Ex. A car can be rented from Continental Rental for $80 per week plus 25 cents for each mile driven. How many miles can you travel if you can spend at most $400 for the week? x = # miles 80 + 0. 25 x ≤ 400 80 + 0. 25 x – 80 ≤ 400 – 80 0. 25 x ≤ 320 0. 25 x ≤ 1280 You can drive up to 1, 280 miles