Question

1. please consider the following inequality:

|x - 4| < 5
what is the next step in solving this inequality?
is this an equivalent inequality?
is |x - 4| < 5 equivalent to the inequality below?
please explain your response in the workspace.
3|x - 4| < 15

Explanation:

First, solve \(|x - 4| < 5\)

Step1: Recall absolute value inequality rule

For \(|A| < B\) (where \(B>0\)), it is equivalent to \(-B < A < B\). So for \(|x - 4| < 5\), we apply this rule.
\(-5 < x - 4 < 5\)

Step2: Solve for \(x\)

Add 4 to all parts of the compound inequality:
\(-5 + 4 < x - 4 + 4 < 5 + 4\)
\(-1 < x < 9\)

Now, check \(|x - 4| < 5\) vs \(3|x - 4| < 15\)

Step1: Simplify \(3|x - 4| < 15\)

Divide both sides of the inequality \(3|x - 4| < 15\) by 3 (since 3 is positive, the inequality sign remains the same).
\(\frac{3|x - 4|}{3} < \frac{15}{3}\)
\(|x - 4| < 5\)

Step2: Conclusion on equivalence

Since simplifying \(3|x - 4| < 15\) (by dividing both sides by 3) gives \(|x - 4| < 5\), and we know how to solve \(|x - 4| < 5\) as above, the two inequalities \(|x - 4| < 5\) and \(3|x - 4| < 15\) are equivalent. This is because multiplying (or dividing) both sides of an inequality by a positive number preserves the inequality relationship, and here we divided \(3|x - 4| < 15\) by 3 (a positive number) to get \(|x - 4| < 5\).

Answer:

s:

• Solution for \(|x - 4| < 5\): \(-1 < x < 9\)
• Next step in solving \(|x - 4| < 5\): Apply the absolute - value inequality rule to get \(-5 < x - 4 < 5\)
• Equivalence of \(|x - 4| < 5\) and \(3|x - 4| < 15\): Yes, they are equivalent because dividing \(3|x - 4| < 15\) by 3 (a positive number) gives \(|x - 4| < 5\)