Question

1. solve $-2(x + 2) < x + 8$. then graph the solution set on a number line.
2. use structure explain how you can solve the inequality $-2x + 4 < 16$ without multiplying or dividing by a negative coefficient.

Explanation:

For Problem 11:

Step1: Expand left-hand side

$-2(x+2) = -2x -4$
Inequality becomes: $-2x -4 < x + 8$

Step2: Isolate x terms to left

Subtract $x$ from both sides:
$-2x -4 -x < x + 8 -x$
$-3x -4 < 8$

Step3: Isolate constant terms to right

Add 4 to both sides:
$-3x -4 +4 < 8 +4$
$-3x < 12$

Step4: Divide by negative coefficient

Divide by $-3$, reverse inequality sign:
$\frac{-3x}{-3} > \frac{12}{-3}$
$x > -4$

Step5: Graph description

On a number line, draw an open circle at $-4$ (since $x$ does not equal $-4$) and shade all values to the right of $-4$.

To solve $-2x + 4 < 16$ without multiplying/dividing by a negative, first isolate the term with $x$ by subtracting the constant from both sides. Then rearrange to keep the positive coefficient of $x$ on one side, avoiding negative division/multiplication entirely.

Answer:

The solution is $x > -4$. To graph: Draw an open circle at $-4$ on a number line, then shade the region to the right of $-4$.

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For Problem 13: