Final answer:
The correct representation of the solution to the inequality 2(x − 2) ≤ 2 is a number line with a closed circle on 3 and shading to the left. Options a, c, and d are correct representations, whereas option b is incorrect.
Step-by-step explanation:
The correct answer is option b - A number line with a closed circle on 3 and shading to the left. When we solve the inequality 2(x − 2) ≤ 2, we first distribute the 2 on the left side obtaining 2x - 4 ≤ 2. Then we add 4 to both sides to get 2x ≤ 6, and finally, we divide both sides by 2 to find the solution x ≤ 3. Representing this on a number line, we would have a closed circle on 3 because 'x is less than or equal to 3' includes the value of 3. The shading would be to the left because we are looking for all values that are less than or equal to 3, not greater than it. Therefore, options a, c, and d are correct representations while option b is incorrect because it implies x is greater than or equal to 3.
To represent the solution of the inequality 2(x - 2) ≤ 2 on a number line, we need to find the values of x that satisfy the inequality.
Step-by-step:
Distribute the 2 on the left side: 2x - 4 ≤ 2
Add 4 to both sides: 2x ≤ 6
Divide both sides by 2 to isolate x: x ≤ 3
Therefore, the solution to the inequality is x ≤ 3. This is represented on a number line with a closed circle on 3 and shading to the left, as option b describes.