To solve the inequality [tex]\(-3 |3 + 3x| + 4 < -23\)[/tex], let’s go through the solution step-by-step:
1. Simplifying the Inequality:
Start by isolating the absolute value expression. To do this, first move the constant term on the right-hand side:
[tex]\[
-3 |3 + 3x| + 4 < -23
\][/tex]
Subtract 4 from both sides:
[tex]\[
-3 |3 + 3x| < -27
\][/tex]
2. Divide by -3:
When dividing or multiplying both sides of an inequality by a negative number, remember to reverse the inequality sign:
[tex]\[
|3 + 3x| > 9
\][/tex]
3. Solve the Absolute Value Inequality:
The inequality [tex]\( |3 + 3x| > 9 \)[/tex] can be split into two separate inequalities:
[tex]\[
3 + 3x > 9 \quad \text{or} \quad 3 + 3x < -9
\][/tex]
4. Solve for x in Each Inequality:
- First Inequality:
[tex]\[
3 + 3x > 9
\][/tex]
Subtract 3 from both sides:
[tex]\[
3x > 6
\][/tex]
Divide by 3:
[tex]\[
x > 2
\][/tex]
- Second Inequality:
[tex]\[
3 + 3x < -9
\][/tex]
Subtract 3 from both sides:
[tex]\[
3x < -12
\][/tex]
Divide by 3:
[tex]\[
x < -4
\][/tex]
5. Combine the Solutions:
The combined solution from both inequalities is:
[tex]\[
x < -4 \quad \text{or} \quad x > 2
\][/tex]
Therefore, the interval notation for the solution is:
[tex]\[
(-\infty, -4) \cup (2, \infty)
\][/tex]
This is the range of values for [tex]\( x \)[/tex] that satisfy the original inequality [tex]\(-3|3 + 3x| + 4 < -23\)[/tex].
