Final answer:
The inequality |2x - 8| < 2 is solved by separating into two cases based on the absolute value. The solution set is all x values between 3 and 5, not including the endpoints. A number line with open circles at 3 and 5, shaded between them, correctly represents the solutions.
Step-by-step explanation:
The question asks which graph represents the solutions to the inequality |2x − 8| < 2. To find the solution, we need to consider two separate cases because the absolute value function defines a distance from zero which can be on either side of the origin.
Case 1: When the expression inside the absolute value is non-negative, we have 2x - 8 < 2. Solving for x, we add 8 to both sides to get 2x < 10, and then divide by 2 to get x < 5.
Case 2: When the expression inside the absolute value is negative, we have -(2x - 8) < 2. This simplifies to -2x + 8 < 2. We subtract 8 from both sides to get -2x < -6, then divide by -2, remembering to reverse the inequality symbol, to get x > 3.
Combining both cases, we need a number line graph that represents all the numbers between 3 and 5 non-inclusively, which means we use open circles at 3 and 5. Therefore, the correct graph is the one with an open circle on 3, shading to the right, and an open circle on 5, shading to the left.