Sure! Let's solve the inequality [tex]\( 3 - (2x - 5) < -4(x + 2) \)[/tex] step by step.
1. Distribute the negative sign inside the parentheses on the left-hand side:
[tex]\[
3 - (2x - 5) \quad \text{becomes} \quad 3 - 2x + 5
\][/tex]
So the inequality now looks like:
[tex]\[
3 - 2x + 5 < -4(x + 2)
\][/tex]
2. Combine like terms on the left-hand side:
[tex]\[
3 + 5 - 2x \quad \text{is} \quad 8 - 2x
\][/tex]
So now we have:
[tex]\[
8 - 2x < -4(x + 2)
\][/tex]
3. Distribute the [tex]\(-4\)[/tex] on the right-hand side:
[tex]\[
-4(x + 2) \quad \text{becomes} \quad -4x - 8
\][/tex]
Now the inequality is:
[tex]\[
8 - 2x < -4x - 8
\][/tex]
4. Move all the [tex]\(x\)[/tex] terms to one side and the constants to the other side. Add [tex]\(4x\)[/tex] to both sides:
[tex]\[
8 - 2x + 4x < -8
\][/tex]
Combine like terms:
[tex]\[
8 + 2x < -8
\][/tex]
5. Subtract 8 from both sides to isolate the [tex]\(x\)[/tex] term:
[tex]\[
2x < -8 - 8
\][/tex]
Simplify the constants:
[tex]\[
2x < -16
\][/tex]
6. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x < \frac{-16}{2}
\][/tex]
Simplify the fraction:
[tex]\[
x < -8
\][/tex]
So the solution to the inequality [tex]\( 3 - (2x - 5) < -4(x + 2) \)[/tex] is:
[tex]\[
x < -8
\][/tex]
