An absolute value equation with a variable on both sides compares a V-shaped graph with a line. Exactly one real solution means the graphs share exactly one point. Compare how steeply the line and the V’s arms rise to find where that can happen. A line above the V’s tip may look like it crosses once, but it crosses the other arm too.
Hints
- Hint 1
The absolute value is the distance from to . That distance is zero at , so the V-shaped graph has its tip there. What is its height?
- Hint 2
A slope tells you how much a graph rises when increases by . To the right of the tip, the V rises by while the line rises by . What happens if they have the same height at the tip?
- Hint 3
For the graphs to meet at the tip, put into the given equation to find . Then use that value in the expression the question asks for; itself is not the answer.
Step-by-step
Make the line meet the V at its tip
Step 1Find the V’s tip
The absolute value measures the distance from to . That distance is zero at , so has its tip at : .
- Step 2
Use the slopes to locate the only possible meeting point
From the tip, the V rises units for each step in either direction. The line rises only units per step to the right; to the left, it falls. So a line through the tip cannot meet the V again. If the line is above the tip, the V eventually catches it on both sides, giving two solutions. Below the tip, it never meets the V. Exactly one solution requires the line to pass through the tip.
- Step 3
Find the line that passes through the tip
Put into the given equation so the line and V have the same height there:
Evaluate both sides:
Subtract from both sides:
- Step 4
Check the single intersection in Desmos
Type , , and . Desmos shows the graphs meet at . The slope comparison shows why there is no second meeting point.
- Step 5
Evaluate the requested expression
Enter on the next Desmos line. It prints . The question asks for this expression, not for , so the value is . Choice C.