Question

1. |3 - x| + 11 < 24

show your work below your answer.
answer: { }

Explanation:

Step1: Isolate the absolute value

Subtract 11 from both sides of the inequality \(|3 - x| + 11 < 24\) to get \(|3 - x| < 24 - 11\).
Simplifying the right side gives \(|3 - x| < 13\).

Step2: Solve the absolute value inequality

Recall that \(|a| < b\) (where \(b>0\)) is equivalent to \(-b < a < b\). So for \(|3 - x| < 13\), we have \(-13 < 3 - x < 13\).

Step3: Solve the left - hand inequality

Subtract 3 from all parts of the compound inequality. First, solve \(-13 < 3 - x\):
Subtract 3 from both sides: \(-13 - 3 < -x\), which simplifies to \(-16 < -x\).
Multiply both sides by - 1 (and reverse the inequality sign) to get \(16 > x\) or \(x < 16\).

Step4: Solve the right - hand inequality

Solve \(3 - x < 13\):
Subtract 3 from both sides: \(-x < 13 - 3\), which simplifies to \(-x < 10\).
Multiply both sides by - 1 (and reverse the inequality sign) to get \(x > - 10\).

Answer:

\(\{x\mid - 10 < x < 16\}\) (or in interval notation \((-10,16)\))