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Precalculus JumpStart

Anthony van Groningen

Exercises 4.8 Exercises

1. Avoiding Common Mistakes.

If \(A\cdot B = 0\) what must \(A\) or \(B\) be equal to? If \(C \cdot D = 1\) can you come to a similiar conclusion?

2. Solving Rational Equations.

Identify the domain of the equation. Then solve each rational equation for \(x\text{.}\) Show all your work as a sequence of equations.
  1. \(\displaystyle \displaystyle \frac{2x+1}{4} = 0\)
  2. \(\displaystyle \displaystyle \frac{4}{2x+1} = 0\)
  3. \(\displaystyle \displaystyle \frac{4x}{2x+1} = 0\)
  4. \(\displaystyle \displaystyle \frac{4x}{2x+1} = 1\)
  5. \(\displaystyle \displaystyle \frac{4}{x+5}-2 = 1\)
  6. \(\displaystyle \displaystyle \frac{5}{2x-8} -3 = -\frac{7}{x-1}\)
  7. \(\displaystyle \displaystyle 3 - \frac{2}{x+1} = \frac{4}{x^2-1}\)
  8. \(\displaystyle \displaystyle \frac{x+1}{x+2}-1 = \frac{x+2}{x-4}\)
Answer.
  1. \(\displaystyle \displaystyle x=-1/2\)
  2. No solutions.
  3. \(\displaystyle \displaystyle x = 0\)
  4. \(\displaystyle \displaystyle x = 1/2\)
  5. \(\displaystyle \displaystyle x = -11/3\)
  6. \(\displaystyle \displaystyle x = 5/2, \, 17/3\)
  7. \(\displaystyle \displaystyle x=5/3\)
  8. \(\displaystyle \displaystyle x=-5, \, 0\)

3. Solving Radical Equations.

Solve each radical equation for \(x\text{.}\) Show all your work as a sequence of equations. Be sure to check your results for extraneous solutions.
  1. \(\displaystyle \displaystyle \sqrt{1-x} = 1\)
  2. \(\displaystyle \displaystyle \sqrt{1-x} = -1\)
  3. \(\displaystyle \displaystyle \sqrt{1-x} - 1 = x\)
  4. \(\displaystyle \displaystyle \sqrt{1-x} +2= x\)
  5. \(\displaystyle \displaystyle \sqrt[3]{1-x} = -1\)
  6. \(\displaystyle \displaystyle \sqrt{x+1} = 2- \sqrt{x+2}\)
Answer.
  1. \(\displaystyle \displaystyle x=0\)
  2. No solutions.
  3. \(\displaystyle \displaystyle x=0\)
  4. No solutions.
  5. \(\displaystyle \displaystyle x = 2\)
  6. \(\displaystyle \displaystyle x=-7/16\)

4. Solving Absolute Value Equations.

Solve each absolute value equation for \(x\text{.}\) Show all your work as a sequence of equations.
  1. \(\displaystyle \displaystyle |2x+1| = 3\)
  2. \(\displaystyle \displaystyle |2x+1| = -3\)
  3. \(\displaystyle \displaystyle |2x-1| = x\)
  4. \(\displaystyle \displaystyle |2x+1| = x\)
Answer.
  1. \(\displaystyle \displaystyle x = -2, \, 1\)
  2. No solutions. The left side of the equation is never negative.
  3. \(\displaystyle \displaystyle x = 1/3, \, 1\)
  4. No solutions. Try solving the equations \(2x+1 = x\) and \(2x+1 = -x\text{.}\) Why do these not lead to solutions?

5. Avoiding Common Mistakes.

Is \(|x+2|\) the same as \(|x|+2\text{?}\) Is \(|2x|\) the same as \(2|x|\text{?}\)
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