To solve the inequality [tex]\(3(4 - 2x) > -2(x + 5)\)[/tex], let's break it down step-by-step:
1. Distribute the constants within the parentheses:
- On the left side, distribute the [tex]\(3\)[/tex]:
[tex]\[3(4 - 2x) = 3 \cdot 4 - 3 \cdot 2x = 12 - 6x\][/tex]
- On the right side, distribute the [tex]\(-2\)[/tex]:
[tex]\[-2(x + 5) = -2 \cdot x - 2 \cdot 5 = -2x - 10\][/tex]
2. Rewrite the inequality with the simplified expressions:
[tex]\[12 - 6x > -2x - 10\][/tex]
3. Combine like terms to isolate the variable [tex]\(x\)[/tex]:
- First, add [tex]\(2x\)[/tex] to both sides of the inequality to move the [tex]\(x\)[/tex]-terms to one side:
[tex]\[12 - 6x + 2x > -2x - 10 + 2x\][/tex]
[tex]\[12 - 4x > -10\][/tex]
- Next, add 10 to both sides to move the constants to the other side:
[tex]\[12 - 4x + 10 > -10 + 10\][/tex]
[tex]\[22 - 4x > 0\][/tex]
4. Isolate the variable [tex]\(x\)[/tex]:
- Subtract 22 from both sides:
[tex]\[22 - 4x - 22 > 0 - 22\][/tex]
[tex]\[-4x > -22\][/tex]
- Divide both sides by [tex]\(-4\)[/tex], remembering to reverse the inequality sign since we are dividing by a negative number:
[tex]\[x < \frac{22}{4}\][/tex]
Simplify the fraction:
[tex]\[x < \frac{11}{2}\][/tex]
So, the inequality simplifies to [tex]\(4x < 22\)[/tex].
Therefore, the equivalent inequality is:
[tex]\[4x < 22\][/tex]
And the correct answer is:
[tex]\[4x < 22\][/tex]
