Solve Rational Equation: Step-by-Step Guide
A rational equation has fractions with x in the bottom. The usual plan is to multiply every term by the least common denominator (LCD) to clear the fractions, then solve the equation you get. Throw out any answer that would make a bottom equal 0.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Find the values that are not allowed. Find every x that makes any denominator 0. These cannot be solutions.\[x \neq r_1, r_2, \dots\]
- Find the LCD. Factor each denominator. The LCD uses each different factor, at its highest power.\[\text{LCD} = \operatorname{lcm}(d_1(x), d_2(x), \ldots)\]
- Multiply every term by the LCD. Multiply every term on both sides by the LCD. This clears all the fractions.\[\text{LCD}\cdot\dfrac{p(x)}{d_1(x)} + \text{LCD}\cdot\dfrac{q(x)}{d_2(x)} = \text{LCD}\cdot r(x)\]
- Solve. Solve the equation you get, using whatever method fits (linear, factoring, or the quadratic formula).\[P(x) = 0\]
- Remove any bad answers and check. Throw out any answer that is a restricted value. Then put the rest into the original equation to check.\[x_0 \notin \{r_1, r_2, \ldots\}\]
Worked example
Solve 1/x + 1/(x + 2) = 3/4.
Step 1: x cannot be 0, and x cannot be -2.
Step 2: The LCD is 4x(x + 2).
Step 3: Multiply each term by 4x(x + 2): 4(x + 2) + 4x = 3x(x + 2).
Step 4: Multiply out: 4x + 8 + 4x = 3x^2 + 6x, so 8x + 8 = 3x^2 + 6x.
Step 5: Move everything to one side: 3x^2 - 2x - 8 = 0. Factor: (3x + 4)(x - 2) = 0, so x = -4/3 or x = 2.
Step 6: Neither answer is 0 or -2, so both are allowed.
Check: For x = 2: 1/2 + 1/4 = 2/4 + 1/4 = 3/4. For x = -4/3: 1/(-4/3) + 1/(2/3) = -3/4 + 3/2 = -3/4 + 6/4 = 3/4. Both work.
Tips and common mistakes
- List the restricted values BEFORE you start. This is how you spot bad answers later.
- Factor each denominator first. Build the LCD from the factored forms.
- Multiply EVERY term by the LCD, even terms that are not fractions.
- Be careful with signs when you multiply through.
- Throw out any answer that is a restricted value. Do not just hope it works out.
- If the equation looks like A/B = C/D, you can cross-multiply: A x D = B x C.
- Clearing fractions can give you a quadratic or something bigger, so be ready to factor or use the quadratic formula.