Solve Absolute Value Inequalities

Conjunction: |ax + b| < c *notice it is <c Means: x is between + c Rewrite the conjunction as -c < ax +b < c This is it’s equivalent compound inequality

Disjunction: |ax +b| > c *notice it is >c Means: not between! Rewrite the disjunction as ax + b < -c or ax + b > c This is it’s equivalent compound inequality

*TIP to help you remember Conjunctions Has a < less than symbol and it requires less work |2x - 1| < 6 Isolate the absolute value, then just drop the bars and put a –6 on the left of the compound inequality and a +6 on the right -6 < 2x - 1 < 6 Then solve like normal Disjunctions Has a > greater than symbol and it requires a greater amount of work |2x - 1| > 6 You must write 2 separate inequalities. Isolate the absolute value. Then for one, just drop the bars and use >6. For the other, drop the bars, flip the inequality sign and use <-6 2x - 1 < -6 or 2x - 1 >6 Then solve like normal

Solving absolute inequalities and graphing: |x - 4| < 3 (less than is betweeness) Means: -3 < x- 4 < 3 Graph: (solve) +4 +4 +4 1< x< 7 0 1 2 3 4 5 6 7 8 9

Solve and graph: *you have to isolate the absolute value bars first! |x + 3| -2 < -1 Means: -1 < x+3 < 1 *now solve Graph: +2 +2 |x + 3| < 1 - 3 - 3 - 3 -4 < x < -2 -5 -4 -3 -2 -1 0 1 2

|x + 9 |> 13 (disjunction) Means: x + 9 < -13 or x + 9 > 13 Solve and graph: |x + 9 |> 13 (disjunction) Means: x + 9 < -13 or x + 9 > 13 -9 -9 -9 -9 x < -22 x > 4 Graph: -25 -20 -15 -10 -5 0 5 10

To write an absolute value inequality from a graph (for Conjunctions) 0 1 2 3 4 5 6 7 8 9 10 1. Write the compound inequality of the graph. (x is between) 2 < x < 8 Find the median (half way between 2 and 8) It’s 5 To find the median, add the two numbers and then divide by 2. 2+8 = 5 2

3. Now rewrite the inequality and subtract the median (5) from each section. 2 - 5 < x - 5 < 8 - 5 Combine like terms (simplify the numbers) and you get -3 < x - 5 < 3 4. Now turn that into it’s corresponding conjunction absolute value inequality |x - 5| < 3 Notice: The median is 5, and the median is 3 units away from either ending number on the number line.

To write an absolute value inequality from a graph (for disjunctions) -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Write the compound inequality of the graph. Find the median. (it’s -1) Subtract the median from all sides (subtract -1 will be adding 1) 4. Write as abs.value inequality. Put x + 1 inside the absolute value bars and positive 5 on the other side of > x < -6 or x > 4 x < -6 or x > 4 +1 x + 1 < - 5 or x+1 > 5 |x+1|>5

Quick rule: Median: Range: |x - median| ( inequality symbol here) range 2 Median: add the two numbers together and divide by 2. Remember to subtract. Watch signs! Range: subtract the two numbers, then divide by 2.