How many real solutions does the equation
have?
A question asking how many solutions a linear equation has calls for simplifying, not necessarily solving for . Distribute and combine like terms, then check what remains if the -terms cancel. You can graph each side in Desmos to see whether they overlap, but the algebra proves whether that overlap is exact. No left does not automatically mean no solutions.
Hints
- Hint 1
The distributive property means multiplying both terms inside the parentheses. Multiply by and by . What happens when you then combine the -terms?
- Hint 2
If the -terms cancel, check the numbers that remain. A true statement, such as , works for every ; a false statement works for none.
Step-by-step
Simplify, then check the graphs
Step 1Distribute the fraction
Multiply by both terms in the parentheses:
- Step 2
Combine like terms
Combine the -terms and the plain numbers:
The -terms cancel, and the numbers cancel too. What remains is a true statement, not an impossible one.
- Step 3
Confirm what the true statement means
Type and on separate Desmos lines. They draw the same horizontal line, as the algebra predicts. Because is true for every real , the equation has infinitely many real solutions. Choice D.