Algebra

(x + 2) / (x - 1) = 3

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Identify the equation to solve

We need to find the value of x that makes this equation true. Note that x cannot equal 1, because that would make the denominator zero.
2

Multiply both sides by denominator

We multiply both sides by (x - 1) to eliminate the fraction. This is called cross-multiplying and lets us work with a simpler equation.
3

Distribute the right side

We use the distributive property to expand 3(x - 1) into 3x - 3. Now we have a simple linear equation with no fractions.
4

Move x terms to one side

Subtract x from both sides and add 3 to both sides to get all x terms on the right and all numbers on the left.
5

Simplify and solve for x

Combine like terms on each side, then divide both sides by 2. We get x = 5/2, which is 2.5. Since 5/2 ≠ 1, this solution is valid.

Understanding this problem

Learning Insight

When a variable appears in a denominator, we have a rational equation. The key idea is that multiplying both sides by the denominator clears the fraction — but we must always check that our answer doesn't make the denominator equal zero, because division by zero is undefined.

Quick Tip

After clearing the fraction, if you end up with a linear equation (highest power of x is 1), there will be exactly one solution. Quadratic-type rational equations can give two solutions, so always check!

Common Mistake

Students often forget to check if the answer makes the denominator zero. For example, if the answer had been x = 1 here, it would have to be thrown out as an invalid (extraneous) solution.