A rational equation has a variable in a denominator. First find any value that makes the denominator zero, because it cannot be a solution. Then enter the original equation as a Desmos regression to find where its sides are equal. You can also clear a shared denominator by hand, but keep the excluded value in view.
Hints
- Hint 1
A denominator of makes a fraction undefined. Set the shared denominator, , equal to first. Which value must you rule out before solving?
- Hint 2
For one equation with one unknown, Desmos can use a regression to find a value that makes both sides equal. Change each to and the equals sign to ; then read the value under PARAMETERS.
Step-by-step
Approach 1: Solve the original equation in Desmos
Step 1Rule out the undefined value
The denominator becomes when . Division by is undefined, so the restriction is . Keep that restriction even if you later clear the fractions.
- Step 2
Find and check the solution
Type . Desmos uses as the number to find, and tells its regression to make the sides equal. Under PARAMETERS, it shows . Since , the value is allowed. The solution is . Choice D.
Approach 2: Clear the shared denominator by hand
Step 1Remove both fractions
For , multiply every term by . That cancels both denominators, while the becomes :
- Step 2
Distribute the 3
Multiply by both terms inside the parentheses:
- Step 3
Combine the constants
Combine and on the right:
- Step 4
Collect the x terms
Subtract from both sides to keep the variable terms together:
- Step 5
Isolate the x term
Subtract from both sides: . Don't lose the minus sign on .
- Step 6
Finish and check the restriction
Divide both sides by : . This is allowed because , so the steps can be reversed to satisfy the original equation. The solution is . Choice D.