Rational Expressions

Grade 10 · algebra · 39 practice problems · read aloud

🔊 Listen to this explanation

Rational Expressions: The Fraction of Algebra

A rational expression is a fraction where the numerator and denominator are polynomials. Think of it as a fancy version of the fractions you already know, but with variables like 'x'. We use them to solve complex problems in engineering, physics, and economics.

🔍 Key Steps to Simplify

  1. Factor Everything: Completely factor both the numerator and the denominator.
  2. State Restrictions: Find the values that make the denominator zero. These are excluded values.
  3. Cancel Common Factors: Cancel any factors that are the same in the top and bottom.
  4. Write Final Answer: Write the simplified expression with its restrictions.

📚 Worked Examples

Example 1: Simplify (3x + 6) / (x² + 4x + 4)

  1. Factor: Numerator: 3(x + 2). Denominator: (x + 2)(x + 2).
  2. Restrictions: (x + 2) = 0, so x ≠ -2.
  3. Cancel: One (x + 2) cancels from top and bottom.
  4. Answer: 3 / (x + 2), where x ≠ -2.

Example 2: Simplify (x² - 9) / (x² - 5x + 6)

  1. Factor: Numerator: (x - 3)(x + 3). Denominator: (x - 2)(x - 3).
  2. Restrictions: x ≠ 2 and x ≠ 3.
  3. Cancel: The (x - 3) factors cancel.
  4. Answer: (x + 3) / (x - 2), where x ≠ 2, 3.

⚠️ Common Mistakes

  • Canceling Terms: You can only cancel factors (things being multiplied), not terms being added/subtracted. (x+1) in the numerator and denominator can cancel, but a standalone 'x' cannot.
  • Forgetting Restrictions: Always state the values that make the original denominator zero. This is a crucial part of the answer.
  • Incorrect Factoring: Solid factoring skills are essential. Practice factoring trinomials and difference of squares.

💡 Tips & Tricks

  • Factor First, Always: Before you do anything else, factor completely.
  • The "Big X": Draw a large X through canceled factors in your work to visually track them.
  • Domain Detective: Find restrictions before canceling, using the original denominator.

🎯 How to Practice

Start with simple expressions that require factoring a GCF or a difference of squares. Then, move to trinomials. Create flashcards with problems on the front and steps on the back. Practice explaining the steps to a friend or family member—teaching is the best way to learn!

Practice problems

6 of the 39, worked through step by step — try them before opening the answer.

1 (3/(x+1)) + (5/(x-3)) = ?

Hint: To add fractions with different denominators, find a common denominator by multiplying the denominators together, then adjust the numerators accordingly.

Show the answer

Answer: (8x-4)/((x+1)(x-3))

  1. Identify the denominators: (x+1) and (x-3)
  2. Find the least common denominator: (x+1)(x-3)
  3. Rewrite each fraction with the common denominator: First fraction: 3/(x+1) = 3(x-3)/((x+1)(x-3)) Second fraction: 5/(x-3) = 5(x+1)/((x+1)(x-3))
  4. Add the numerators: 3(x-3) + 5(x+1) = 3x-9 + 5x+5 = 8x-4
  5. Write the result: (8x-4)/((x+1)(x-3))

The answer is (8x-4)/((x+1)(x-3)).

2 (9/(x+2)) + (7/(x-3)) = ?

Hint: To add fractions with different denominators, find a common denominator by multiplying the denominators together. Then rewrite each fraction with this common denominator before adding.

Show the answer

Answer: (16x-13)/((x+2)(x-3))

  1. Identify the denominators: (x+2) and (x-3)
  2. Find the least common denominator: (x+2)(x-3)
  3. Rewrite the first fraction: 9/(x+2) = 9(x-3)/((x+2)(x-3))
  4. Rewrite the second fraction: 7/(x-3) = 7(x+2)/((x+2)(x-3))
  5. Add the numerators: 9(x-3) + 7(x+2) = 9x - 27 + 7x + 14 = 16x - 13
  6. Write the result: (16x-13)/((x+2)(x-3))

The answer is (16x-13)/((x+2)(x-3)).

3 (7/(x+2)) + (9/(x-4)) = ?

Hint: To add fractions with different denominators, find a common denominator first. For example, when adding 2/(a+1) + 3/(a-2), the common denominator would be (a+1)(a-2).

Show the answer

Answer: (16x-10)/((x+2)(x-4))

  1. Identify the denominators: (x+2) and (x-4)
  2. Find the least common denominator: (x+2)(x-4)
  3. Rewrite each fraction with the common denominator: 7/(x+2) = 7(x-4)/((x+2)(x-4)) 9/(x-4) = 9(x+2)/((x+2)(x-4))
  4. Add the numerators: 7(x-4) + 9(x+2) = 7x-28 + 9x+18 = 16x-10
  5. Write the result: (16x-10)/((x+2)(x-4))

The answer is (16x-10)/((x+2)(x-4)).

4 (4/(x+2)) + (6/(x-4)) = ?

Hint: When adding fractions with different denominators, find a common denominator by multiplying the denominators together. Then adjust the numerators accordingly before combining.

Show the answer

Answer: (10x-4)/((x+2)(x-4))

  1. Identify the denominators: (x+2) and (x-4)
  2. Find the common denominator: (x+2)(x-4)
  3. Rewrite each fraction with the common denominator: First fraction: (4/(x+2)) = (4(x-4))/((x+2)(x-4)) = (4x-16)/((x+2)(x-4)) Second fraction: (6/(x-4)) = (6(x+2))/((x+2)(x-4)) = (6x+12)/((x+2)(x-4))
  4. Add the numerators: (4x-16) + (6x+12) = 10x-4
  5. Write the result: (10x-4)/((x+2)(x-4)) The simplified expression is (10x-4)/((x+2)(x-4))

5 (5/(x+3)) + (7/(x-5)) = ?

Hint: When adding fractions with different denominators, find a common denominator by multiplying the denominators together. Then rewrite each fraction with this common denominator before adding.

Show the answer

Answer: (12x-4)/((x+3)(x-5))

  1. Identify the denominators: (x+3) and (x-5)
  2. The common denominator is (x+3)(x-5)
  3. Rewrite the first fraction: 5/(x+3) = 5(x-5)/[(x+3)(x-5)]
  4. Rewrite the second fraction: 7/(x-5) = 7(x+3)/[(x+3)(x-5)]
  5. Add the numerators: 5(x-5) + 7(x+3) = 5x - 25 + 7x + 21 = 12x - 4
  6. Write the result: (12x-4)/[(x+3)(x-5)]

The answer is (12x-4)/((x+3)(x-5))

6 (6/(x+4)) + (8/(x-2)) = ?

Hint: When adding fractions with different denominators, find a common denominator by multiplying the denominators together. Then adjust the numerators accordingly.

Show the answer

Answer: (14x+20)/((x+4)(x-2))

  1. Identify the denominators: (x+4) and (x-2)
  2. Find the common denominator: (x+4)(x-2)
  3. Rewrite each fraction with the common denominator: 6/(x+4) = 6(x-2)/((x+4)(x-2)) 8/(x-2) = 8(x+4)/((x+4)(x-2))
  4. Add the numerators: 6(x-2) + 8(x+4) = 6x - 12 + 8x + 32 = 14x + 20
  5. Write the result: (14x+20)/((x+4)(x-2))

The answer is (14x+20)/((x+4)(x-2)).

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