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7.2 – Rational Expressions

Notes

Radicals & Rational Numbers
  • A rational number is a quotient, or ratio, of two integers.
  • Rational number examples
    • Ex 1. 12=\frac{12}{1}
    • Ex 2. -5=\frac{-5}{1}
    • Ex 3. \sqrt{\frac{16}{9}}=\frac{4}{3}
  • A rational expression contains numbers or variables separated by a plus or minus sign.
  • Rational expression ex
    • Ex 1. \frac{12-4}{3-12}
    • Ex 2. \frac{12-x}{3x-6}

Notes

Rational Expressions
  • Equal to Zero
    • A rational expression is equal to zero when its numerator is zero.
  • Undefined
    • A rational expression is undefined whenever its denominator is zero.

Examples

  • Ex 1. For what value of x is a rational expression undefined?
    • Answer: When the denominator is equal to 0
  • Ex 2. For what value of x is a rational expression below equal to zero?
    • Answer: When the numerator is equal to 0
  • Ex 3. What is the value of the rational expression below when x is equal to 3?
  • \frac{12-x}{x-6}
    • Substitute 3 for x: \frac{12-3}{3-6}
    • Simplify: \frac{9}{-3}
      • Answer: -3
  • Ex 4. For what value of x is the rational expression below equal to zero?
  • \frac{x-4}{x-6}
    • Setup: x – 4 = 0
    • Do the inverse operation (add 4 to both sides)
      • Answer: x = 4
  • Ex 5. For what value of x is the rational expression below undefined?
  • \frac{x-4}{4+x}
    • Setup: 4 + x = 0
    • Do the inverse operation (subtract 4 from both sides)
      • Answer: x = -4
  • Ex 6. For what values of x is the rational expression below undefined?
  • \frac{x-7}{x^{2}-7x-8}
    • Setup: Set the denominator equal to zero and solve.
    • Try factoring (FOIL): x^{2}-7x-8=0
    • (x-8)(x+1) works!
    • Set the first factor equal to zero: x-8=0
    • Add 8 to both sides: x = 8
    • Set the second factor equal to zero: x+1=0
      • Subtract 1 from both sides.  Answer: x = -1
  • Ex 7. For what values of x is the rational expression below undefined?
  • \frac{x+5}{3x^{2}-3}
    • Setup: Set the denominator equal to zero and solve.
    • Factor out a 3: 3(x^{2}-1)=0
    • Add 1 to both sides: x^{2}=1
    • Set the variable factor equal to zero: x^{2}=1
    • Take the square root of both sides: \sqrt{x^{2}}=\sqrt{1}
      • Answer: x\pm 1

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