Which set describes all real numbers that satisfy the inequality ?
An absolute value inequality with on both sides compares a V-shaped graph with a line. Graph the two sides in Desmos and find where they meet, then look for where the left graph is higher. If a meeting point appears as a decimal, solve that arm of the V for an exact cutoff. A strict leaves the meeting points out.
Hints
- Hint 1
An intersection is where two graphs have the same height. Here, intersections mark cutoffs, not values that satisfy . Which graph must be higher for the inequality to hold?
- Hint 2
Each arm of the V is a straight line. On the left arm, is negative, so absolute value changes to . Use that to find the exact left cutoff.
- Hint 3
The sign means strictly greater. At either cutoff the sides are equal, so neither cutoff counts. Which regions of the graph have the V above the line?
Step-by-step
Graph the sides, then find exact cutoffs
Step 1Locate where the two sides meet
Type and on separate Desmos lines. Click the intersections, where the graphs meet: Desmos shows approximately and . Their -coordinates are equality cutoffs, not yet the values that satisfy . Each arm of the V is straight, so it cannot cross the line twice.
- Step 2
Find the exact left cutoff
The left intersection has negative , so is negative there. Absolute value changes its sign, giving the cutoff equation:
Subtract from both sides:
Subtract from both sides:
Divide by :
- Step 3
Find the exact right cutoff
At the right intersection, is positive, so the bars leave it unchanged. Set the two sides equal:
Add to both sides:
Subtract from both sides:
- Step 4
Keep the regions where the left side is greater
Add on a third Desmos line. It shades the outer regions, where the V is above the line. For a strict inequality, keep where the left graph is higher, but leave out points where the graphs meet. So all real solutions are or . Choice A.