Sure, let's solve the given inequality step-by-step:
Given inequality:
[tex]\[ 2|x-1| + 5 < 13 \][/tex]
Step 1: Subtract 5 from both sides of the inequality:
[tex]\[ 2|x-1| + 5 - 5 < 13 - 5 \][/tex]
[tex]\[ 2|x-1| < 8 \][/tex]
Step 2: Divide both sides of the inequality by 2:
[tex]\[ \frac{2|x-1|}{2} < \frac{8}{2} \][/tex]
[tex]\[ |x-1| < 4 \][/tex]
Step 3: Consider the property of absolute values. The expression [tex]\(|x-1| < 4\)[/tex] means that [tex]\( x-1 \)[/tex] lies between [tex]\(-4\)[/tex] and [tex]\( 4 \)[/tex]:
[tex]\[ -4 < x-1 < 4 \][/tex]
Step 4: Add 1 to all parts of the inequality to isolate [tex]\( x \)[/tex]:
[tex]\[ -4 + 1 < x-1 + 1 < 4 + 1 \][/tex]
[tex]\[ -3 < x < 5 \][/tex]
Thus, the solution for the inequality is:
[tex]\[ -3 < x < 5 \][/tex]
