Question

1. which of the following is equivalent to $|x - 7| < 1$?

choose 3 answers that apply.
$x - 7 > 1$ or $x - 7 < -1$
$5|x - 7| - 4 < 1$
$-1 < x - 7 < 1$
$-|x - 7| > -1$
$2|x - 7| < 4$

Explanation:

Step1: Analyze the first option

The inequality \(|x - 7| < 1\) represents the distance between \(x\) and \(7\) is less than \(1\). The solution to \(|a| < b\) (where \(b>0\)) is \(-b < a < b\), not \(a > b\) or \(a < -b\). So \(x - 7>1\) or \(x - 7 < - 1\) is the solution for \(|x - 7|>1\), not \(|x - 7| < 1\). So this option is incorrect.

Step2: Analyze the second option

Start with \(5|x - 7|-4 < 1\). Add \(4\) to both sides: \(5|x - 7| < 1 + 4=5\). Then divide both sides by \(5\): \(|x - 7| < 1\), which is the original inequality. So this option is correct.

Step3: Analyze the third option

By the definition of absolute value inequality, if \(|a| < b\) (\(b>0\)), then \(-b < a < b\). Here \(a=x - 7\) and \(b = 1\), so \(-1

Step4: Analyze the fourth option

Start with \(-|x - 7|>-1\). Multiply both sides by \(- 1\) (and reverse the inequality sign): \(|x - 7| < 1\) (because when we multiply or divide an inequality by a negative number, the inequality sign flips). So this option is correct.

Step5: Analyze the fifth option

Start with \(2|x - 7| < 4\). Divide both sides by \(2\): \(|x - 7| < 2\), which is not equivalent to \(|x - 7| < 1\). So this option is incorrect.

Answer:

The correct options are:

• \(5|x - 7|-4 < 1\)
• \(-1 < x - 7 < 1\)
• \(-|x - 7| > - 1\)