To solve the inequality [tex]\( 2(4x - 3) \leq -3(3x) + 5x \)[/tex], let's go through this step by step:
1. Distribute the constants inside the parentheses:
- For [tex]\( 2(4x - 3) \)[/tex]:
[tex]\[
2 \cdot 4x - 2 \cdot 3 = 8x - 6
\][/tex]
- For [tex]\( -3(3x) \)[/tex]:
[tex]\[
-3 \cdot 3x = -9x
\][/tex]
2. Substitute these into the inequality:
[tex]\[
8x - 6 \leq -9x + 5x
\][/tex]
3. Combine like terms on the right-hand side:
[tex]\[
-9x + 5x = -4x
\][/tex]
So the inequality now is:
[tex]\[
8x - 6 \leq -4x
\][/tex]
4. Move all the [tex]\( x \)[/tex] terms to one side and constants to the other side:
Add [tex]\( 4x \)[/tex] to both sides:
[tex]\[
8x + 4x - 6 \leq 0
\][/tex]
Simplify:
[tex]\[
12x - 6 \leq 0
\][/tex]
5. Isolate the [tex]\( x \)[/tex] term:
Add 6 to both sides:
[tex]\[
12x \leq 6
\][/tex]
Divide by 12:
[tex]\[
x \leq \frac{6}{12}
\][/tex]
Simplify:
[tex]\[
x \leq \frac{1}{2}
\][/tex]
So the solution to the inequality is:
[tex]\[
x \leq \frac{1}{2}
\][/tex]
In interval notation, this is:
[tex]\[
(-\infty, 0.5]
\][/tex]
Thus, the correct answer is:
[tex]\[
(-\infty, 0.5]
\][/tex]
