For an absolute value equation with a variable on both sides, graph each side in Desmos and count where the graphs meet. Each intersection gives one real solution. Watch for a line that touches the curve without crossing it: the two sides are equal there too.
Hints
- Hint 1
An absolute value is never negative. Since it equals , what must be true about at every real solution?
- Hint 2
An intersection is a point where two graphs have the same height. Graph each side of the equation separately so you can see where their heights match.
- Hint 3
A line can touch a curve without crossing it. The sides are still equal at that point, so check for touches as well as crossings.
Step-by-step
Graph both sides and count
Step 1Rule out negative values of
The left side is an absolute value, so it can't be negative. It equals , so the right side must be nonnegative: . No point with negative can solve the equation.
- Step 2
Graph the original sides
Type and on separate Desmos lines. An intersection is where the graphs share a height, so its -coordinate solves the original equation. Keep the bars: removing them would change the graph.
- Step 3
Count every shared point
Click the shared points near , , and . The outer points are crossings; the graphs touch at . A touch counts as an intersection even without a crossing. There are three distinct -values, so the equation has three real solutions. Choice C.