Solving Absolute Value Inequalities
Solving Absolute Value Inequalities 1. ax+b 0 Becomes an “and” problem Changes to: –c < ax+b < c 2. ax+b > c, where c > 0 Becomes an “or” problem Changes to: ax+b > c or ax+b < -c
To SOLVE an absolute value inequality : ● Step 1: ISOLATE the absolute value expression. ● Step 2: Rewrite the inequality as an intersection or a union. ● If you have a you are working with an intersection or an ‘and’ statement. Remember: “Less thand” ● If you have a you are working with a union or an ‘or’ statement. Remember: “Greator” When writing the 2 nd part of the inequality, you must negate one value and reverse the direction of the inequality sign. ● Step 3: Solve the compound inequality.
Example 1: ● |2x + 1| > 7 ● 2x x 6 ● x 3 This is an ‘or’ statement. (Greator). Rewrite as a union. Reverse the inequality sign and negate the right side value of one of the inequalities. Solve each inequality. Graph the solution. 3 -4
Example 2: ● |x -5|< 3 ● - 3< x -5< 3 ● 2 < x < 8 This is an ‘and’ statement. (Less thand). Rewrite as an intersection. The negated value moves to the left and use the same inequality sign for both parts of the inequality (either < or ≤). Solve each inequality. Graph the solution. 8 2
Solve & graph. Becomes an “and” problem
Solve & graph Isolate the absolute value expression first. Becomes an “or” problem
Solve. ● 1) |y – 3| – 5 > - 4 ● y 4 ● 2) -3|p + 2| +7 ≥ -11 ● -8 ≤ p ≤ 4
Homework: p. 56 #54-63 multiples of 3, all
Writing Absolute Value Inequalities
Write a range as an absolute value. ● Calculate the mean of the extremes (range). ● Find the tolerance (subtract the mean from the upper extreme). ● Write as an absolute value inequality.
Example: A manufacturer of sleeping bags suggests that one model is best suited for temperatures between 30°F and 60°F, inclusive. Write an absolute value inequality for this temperature range. Let t be the actual temperature. Calculate the mean: Find the tolerance: Write as an absolute value inequality:
Homework: p. 57 #75, 77, and 80