To solve the inequality [tex]\( -2|3d - 5| \leq 10 \)[/tex], let's break it down step-by-step:
1. Isolate the Absolute Value Expression:
First, isolate the absolute value expression by dividing both sides of the inequality by -2. Remember that dividing by a negative number reverses the inequality sign.
[tex]\[
|3d - 5| \geq -\frac{10}{-2}
\][/tex]
Simplifying this, we get:
[tex]\[
|3d - 5| \geq 5
\][/tex]
2. Understand the Meaning of the Absolute Value Inequality:
The inequality [tex]\( |3d - 5| \geq 5 \)[/tex] means that the expression [tex]\( 3d - 5 \)[/tex] is either less than or equal to -5, or greater than or equal to 5. This leads to two separate inequalities.
[tex]\[
3d - 5 \leq -5 \quad \text{or} \quad 3d - 5 \geq 5
\][/tex]
3. Solve Each Inequality Separately:
- For [tex]\( 3d - 5 \leq -5 \)[/tex]:
[tex]\[
3d - 5 \leq -5
\][/tex]
Add 5 to both sides:
[tex]\[
3d \leq 0
\][/tex]
Divide by 3:
[tex]\[
d \leq 0
\][/tex]
- For [tex]\( 3d - 5 \geq 5 \)[/tex]:
[tex]\[
3d - 5 \geq 5
\][/tex]
Add 5 to both sides:
[tex]\[
3d \geq 10
\][/tex]
Divide by 3:
[tex]\[
d \geq \frac{10}{3}
\][/tex]
4. Combine the Solutions:
The solutions to the inequalities are:
[tex]\[
d \leq 0 \quad \text{or} \quad d \geq \frac{10}{3}
\][/tex]
5. Express the Final Solution:
In interval notation, the solution is:
[tex]\[
(-\infty, 0] \cup \left[\frac{10}{3}, \infty\right)
\][/tex]
Thus, the solution to the inequality [tex]\( -2|3d - 5| \leq 10 \)[/tex] is [tex]\( (-\infty, 0] \cup \left[\frac{10}{3}, \infty\right) \)[/tex].
