Which of the following describes all real values of that satisfy the inequality
An absolute-value inequality compares a V-shaped graph with another graph. When the variable appears on both sides, graph each original side in Desmos and find every crossing. Then compare which graph is higher in each region. Don’t assume the answer lies between the crossings: the V may be higher outside them.
Hints
- Hint 1
An intersection is where two graphs have the same height. Those crossings mark where the inequality might change from true to false. Graph and , and look for both crossings.
- Hint 2
Desmos shows a crossing near , but that decimal is rounded. The options suggest as an exact cutoff. What does each side of the inequality equal when ?
- Hint 3
The symbol means the left side can be equal to or greater than the right side. Compare the graphs to the left, between, and to the right of the crossings. Do the crossings themselves count?
Step-by-step
Graph both sides and compare
Step 1Find where the graphs cross
Type and in Desmos. Click both intersections: their -coordinates are about and . An intersection is where the two sides are equal, so these values divide the graph into regions to check.
- Step 2
Check the exact fractional cutoff
The first crossing is rounded on the graph, so test the proposed cutoff . Type each side with . Desmos displays about for each. Tap the fraction button on each line to see for both, confirming that is an exact crossing.
- Step 3
Keep the regions where the inequality holds
On the graph, the V is higher than the line to the left of and to the right of ; it’s lower between them. The sign keeps the regions where the V is on or above the line. Equality includes both crossings, so all real solutions are or . Choice D.