Final answer:
The values of x that satisfy the inequality |x-2| ≤ x+1 are -∞ < x ≤ 1.5 and x ≥ 2. To sketch the inequality's graph, plot the points 1.5 and 2 on a number line and shade to the left of 1.5 and right of 2, including point 2.
Step-by-step explanation:
To find all values of x that satisfy the inequality |x-2| ≤ x+1, we need to consider two cases, since the absolute value of a number is the distance of that number from zero on the number line, and is always non-negative:
Case 1: x - 2 ≤ x + 1, when x ≥ 2, which simplifies to -2 ≤ 1, a true statement. So, for x ≥ 2, all values of x are solutions.
Case 2: -(x - 2) ≤ x + 1, when x < 2, which simplifies to 3 ≥ 2x, or x ≤ 1.5.
Combining both cases, the solution to the inequality is -∞ < x ≤ 1.5 and x ≥ 2.
To sketch the graph of this inequality, we would:
Draw a number line.
Plot and label the critical points 1.5 and 2.
Shade the region to the left of 1.5 and to the right of 2, including the point 2 since x can equal 2.
The shaded area would represent the range of values that x can take.