Solving Rational Equations

Solve equations with rational expressions and check solutions. Grade 11 Math 20-1.

Lesson 6.5 of Rational Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Expressions vs. Equations. Understand the fundamental difference between an expression (no equals sign — simplify only) and an equation (equals sign — solve for a value), with a worked comparison example.
• Solving Rational Equations. Learn the five-step procedure: factor, state NPVs, multiply by the LCD to clear fractions, solve the resulting equation, and reject any extraneous roots — applied across six progressively complex examples.
• Working Backwards — Finding Unknown Values. Use the structure of a rational equation to find unknown parameters: side-by-side comparison of simplifying an expression vs. solving an equation, and determining an unknown coefficient from given solutions.

Inside the lesson — a free preview

The Key Difference

Expression vs. Equation — At a Glance. The single most important distinction in algebra is whether you have an expression or an equation. An expression has no equals sign — you can only simplify it. An equation has an equals sign — you can perform the same operation on both sides to find a specific value for the variable.

Investigate — Compare Side by Side

The two problems below look similar, but they require completely different techniques.

Problem 1 — Simplify the Expression 2x - 3y + 3(x + 2y) - 4x. No equals sign — this is an expression. We can only combine like terms.

Step 1 — Distribute the 3.

3(x + 2y) = 3x + 6y

Step 2 — Start with the given expression.

2x - 3y + 3(x + 2y) - 4x

Step 3 — Substitute back and group like terms.

2x - 3y + 3x + 6y - 4x

x terms: 2x + 3x - 4x = x

y terms: -3y + 6y = 3y

Simplified expression: x + 3y

Problem 2 — Solve the Equation 2x + 3(3y - 2) = 4(x + 3) + 9y. Has an equals sign — this is an equation. We can perform the same operation on both sides.

Step 1 — Distribute on both sides.

3(3y - 2) = 9y - 6

4(x + 3) = 4x + 12

Substituting: 2x + 9y - 6 = 4x + 12 + 9y

Step 2 — Subtract 9y from both sides.

2x - 6 = 4x + 12

Step 3 — Subtract 2x from both sides.

-6 = 2x + 12

Step 4 — Subtract 12 from both sides.

-18 = 2x

Step 5 — Divide both sides by 2.

Solution: x = -9

The Takeaway. With an expression, you can only perform the indicated operations — combining like terms and simplifying. With an equation, you can perform the same operations to both sides to isolate the variable and find a solution.

Worked example: a — Constant Denominators

Solve (x)/(2) + 3 = 2 + (3x)/(4).

Example 1a — (x)/(2) + 3 = 2 + (3x)/(4). Step 1 — Find NPVs.

The denominators are 2 and 4 — both constants with no variables. There are no NPVs.

Step 2 — Find the LCD.

The denominators are 2 and 4. Since 4 = 2^2, the LCD = 4.

Step 3 — Multiply every term on both sides by 4.

4 · (x)/(2) + 4 · 3 = 4 · 2 + 4 · (3x)/(4)

2x + 12 = 8 + 3x

Step 4 — Solve for x.

Subtract 2x from both sides: 12 = 8 + x

Subtract 8 from both sides: 4 = x

Step 5 — Check for extraneous roots.

There were no NPVs, so x = 4 is valid.

Answer: x = 4

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