Question

Kim is solving the absolute value inequality |2x – 1| + 3 < 6.

|2x – 1| + 3 < 6
|2x – 1| + 3 – 3 < 6 – 3
|2x – 1| < 3
–3 < 2x – 1 or 2x – 1 < 3
–2 < 2x or 2x < 4
–1 < x or x < 2

Where did Kim go wrong? Explain your reasoning.
A. Kim should have isolated the absolute value before writing the compound inequality.
B. Kim should have used –3 > 2x –1 rather than –3 < 2x –1.
C. Kim should have used "and" instead of "or" in her compound inequality.
D. Kim should have reversed the inequality symbol when solving 2x > –2.

Answer 1

Absolute values creates an interval implied by \(|\bullet|\). The conditions are that \(\bullet\) be less than something AND greater than -something.

Answer 2

So,
$$
|2x – 1| < 3\\
\implies -3 < 2x-1 < 3\\
2x-1 < 3 \text{ AND }
-3 < 2x-1
$$

Here 2x-1 is \(\bullet\) and 3 is "something"

Answer 3

In more mathematical terms,

|blah blah blah| < \(something\) means blah blah blah < \(something\) AND \(-something \) < blah blah blah