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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