Question

solve: 2|4x+6|-13 ≤ -7
give your answer as an interval if no solutions exist, enter no solutions

Answer 1

To solve the inequality \( 2|4x + 6| - 13 \leq -7 \), we will first simplify the inequality.

1. Add 13 to both sides:
\[
2|4x + 6| \leq -7 + 13
\]
\[
2|4x + 6| \leq 6
\]

2. Divide both sides by 2:
\[
|4x + 6| \leq 3
\]

Now we can interpret the absolute value inequality:
\[
-3 \leq 4x + 6 \leq 3
\]

3. We'll break this into two separate inequalities:

- For the left side:
\[
-3 \leq 4x + 6
\]
Subtract 6 from both sides:
\[
-3 - 6 \leq 4x
\]
\[
-9 \leq 4x
\]
Divide by 4:
\[
-\frac{9}{4} \leq x
\]
or
\[
x \geq -\frac{9}{4}
\]

- For the right side:
\[
4x + 6 \leq 3
\]
Subtract 6 from both sides:
\[
4x \leq 3 - 6
\]
\[
4x \leq -3
\]
Divide by 4:
\[
x \leq -\frac{3}{4}
\]

Now, we combine the two parts:
\[
-\frac{9}{4} \leq x \leq -\frac{3}{4}
\]

4. In interval notation, the solution is:
\[
\left[-\frac{9}{4}, -\frac{3}{4}\right]
\]

Thus, the solution to the inequality \( 2|4x + 6| - 13 \leq -7 \) is:
\[
\left[-\frac{9}{4}, -\frac{3}{4}\right]
\]