Question

1. solve the inequality: $-7(8 + 2x) + 6x > -112$

a. $x < -10$
b. $x < -38$
c. $x < 0$
d. $x < 7$

Explanation:

Step1: Distribute -7

First, distribute -7 into the parentheses: $-7(8 + 2x) = -56 - 14x$. So the inequality becomes $-56 - 14x + 6x > -112$.

Step2: Combine like terms

Combine the x - terms: $-14x + 6x = -8x$. Now the inequality is $-56 - 8x > -112$.

Step3: Add 56 to both sides

Add 56 to both sides of the inequality: $-8x > -112 + 56$. Simplify the right - hand side: $-8x > -56$.

Step4: Divide by -8 (and reverse inequality)

Divide both sides of the inequality by -8. Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So we have $x < \frac{-56}{-8}$. Simplify the right - hand side: $x < 7$. Wait, no, wait. Wait, let's re - do step 3. Wait, $-112+56=-56$, so $-8x > - 56$. Dividing both sides by -8: $x < \frac{-56}{-8}=7$? Wait, no, that's not matching the options. Wait, I must have made a mistake. Let's start over.

Wait, original inequality: $-7(8 + 2x)+6x > - 112$

Step 1: Distribute -7: $-56-14x + 6x > - 112$

Step 2: Combine like terms: $-56-8x > - 112$

Step 3: Add 56 to both sides: $-8x > - 112 + 56=-56$

Step 4: Divide both sides by -8: When dividing by a negative number, the inequality sign flips. So $x < \frac{-56}{-8}=7$? But option D is $x < 7$. Wait, but let's check again. Wait, maybe I made a mistake in the sign. Wait, $-7(8 + 2x)=-56-14x$, then $-56-14x + 6x=-56 - 8x$. So the inequality is $-56 - 8x > - 112$. Add 56 to both sides: $-8x > - 56$. Divide by -8: $x < 7$. So the answer should be D. $x < 7$? But let's check with x = 0: Left - hand side: $-7(8 + 0)+0=-56 > - 112$, which is true. x = 7: $-7(8 + 14)+42=-7\times22 + 42=-154 + 42=-112$, but the inequality is $> - 112$, so x must be less than 7. So the correct answer is D. $x < 7$? Wait, but let's check the options again. Option D is $x < 7$. So that's correct.

Wait, but maybe I made a mistake in the calculation. Let's try x = 6: $-7(8 + 12)+36=-7\times20 + 36=-140 + 36=-104 > - 112$, which is true. x = 7: $-7(8 + 14)+42=-154 + 42=-112$, which is not greater than - 112. x = 8: $-7(8+16)+48=-7\times24 + 48=-168 + 48=-120 < - 112$, which is false. So yes, when x < 7, the inequality holds. So the answer is D.

Wait, but the options are A. $x < - 10$, B. $x < - 38$, C. $x < 0$, D. $x < 7$. So the correct answer is D. $x < 7$.

Answer:

D. \( x < 7 \)