Question

Which graph represents the solution set for –4(1 – x) ≤ –12 + 2x?

Answer 1

To find the graph that represents the solution set for the inequality −4(1 − x) ≤ −12 + 2x, simplify the inequality to get x ≤ −4, which is represented graphically by a number line shaded from −4 to negative infinity, including the point −4.

Step-by-step explanation:

To solve the inequality −4(1 − x) ≤ −12 + 2x, first we simplify both sides of the inequality:

• Multiply the −4 inside the parentheses: −4 + 4x ≤ −12 + 2x.
• Combine like terms on both sides: 4x − 2x ≤ −12 + 4.
• Simplify the inequality: 2x ≤ −8.
• Finally, divide both sides by 2 to isolate x: x ≤ −4.

The solution to the inequality is all real numbers less than or equal to −4. Graphically, this is represented by a shaded region on the number line that extends from negative infinity to −4, including the point −4 (which would be a closed circle on the graph).

Answer 2

`-4(1-x)\leq-12+2x\qquad\text{use distributive property}\\\\(-4)(1)+(-4)(-x)\leq-12+2x\\\\-4+4x\leq-12+2x\qquad\text{add 4 to both sides}\\\\4x\leq-8+2x\qquad\text{subtract 2x from both sides}\\\\2x\leq-8\qquad\text{divide both sides by 2}\\\\\boxed{x\leq-4}`