Let's solve the inequality [tex]\(3|m+7| + 9 < 30\)[/tex] step by step.
Step 1: Subtract 9 from both sides of the inequality to isolate the absolute value expression.
[tex]\[
3|m+7| + 9 < 30
\][/tex]
[tex]\[
3|m+7| < 21
\][/tex]
Step 2: Divide both sides of the inequality by 3 to further isolate the absolute value.
[tex]\[
|m+7| < 7
\][/tex]
Step 3: The inequality [tex]\(|m+7| < 7\)[/tex] implies that the expression inside the absolute value is within 7 units of 0. This can be written as two separate inequalities:
[tex]\[
-7 < m + 7 < 7
\][/tex]
Step 4: Subtract 7 from all parts of the compound inequality to solve for [tex]\(m\)[/tex]:
[tex]\[
-7 - 7 < m + 7 - 7 < 7 - 7
\][/tex]
[tex]\[
-14 < m < 0
\][/tex]
So the solution to the inequality [tex]\(3|m+7| + 9 < 30\)[/tex] is:
[tex]\[
-14 < m < 0
\][/tex]
This means [tex]\(m\)[/tex] must be a value that lies strictly between -14 and 0.
