ACT Preparation / Unit 9 ACT 0823 worked example, 60 to 90 seconds
Solving an Absolute Value Inequality
The idea
To solve an absolute value inequality, work in four steps. First, isolate the absolute value on one side. Second, decide its type: less than means one interval, between; greater than means two pieces, outside. Third, write the matching compound inequality without the bars and solve it. Fourth, test a value from each region.
In practice
Worked example: 2|x + 1| − 3 ≤ 7. Step one: 2|x + 1| ≤ 10, so |x + 1| ≤ 5. Step two: less than, so one interval. Step three: −5 ≤ x + 1 ≤ 5, so −6 ≤ x ≤ 4. Step four: x = 0 gives 2 − 3 = −1 ≤ 7, true; x = 5 gives 12 − 3 = 9 ≤ 7, false.
Where people slip
Isolate before splitting. Splitting 2|x + 1| − 3 ≤ 7 while the 2 and the 3 still sit outside the bars produces the wrong bounds.
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