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Chapter 8: Problem 12

In the following exercises, evaluate the rational expression for the given values. \(\frac{b^{2}+2}{b^{2}-3 b-4}\) (a) \(b=0\) (b) \(b=2\) c) \(b=-2\)

Short Answer

Expert verified
a) -1/2, b) -1, c) 1

Step by step solution

01

Substitute and Simplify for b=0

First, substitute the value of b into the expression. For b=0, the expression becomes \(\frac{0^{2}+2}{0^{2}-3 \cdot 0-4}\). Simplify the expression: \(\frac{2}{-4} = -\frac{1}{2}\).
02

Substitute and Simplify for b=2

Next, substitute b=2. The expression becomes \(\frac{(2)^{2}+2}{(2)^{2}-3 \cdot 2-{4}} = \frac{4+2}{4-6-4} = \frac{6}{-6} = -1\).
03

Substitute and Simplify for b=-2

Finally, substitute b=-2. The expression becomes \(\frac{(-2)^{2}+2}{(-2)^{2}-3 \cdot (-2)-4} =\frac{4+2}{4+6-4} = \frac{6}{6} = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method in algebra
To solve rational expressions using the substitution method in algebra, you replace the variable with given values. This technique simplifies the problem into a straightforward numerical calculation.

First, identify the variable and the value you need to substitute. Take it step-by-step:
  • Replace the variable with the given number.
  • Perform the arithmetic operations inside the expression.
  • Simplify the result to its simplest form.

In our example, we evaluated \(\frac{b^{2}+2}{b^{2}-3 b-4}\) for different values of \(b\): 0, 2, and -2. Each substitution turns the expression into a simple fraction to be simplified. This method is very useful for checking specific scenarios or solving multiple-choice problems.
simplifying rational expressions
Simplifying rational expressions involves reducing them to their simplest form, making them easier to work with. Follow these steps:

  • Factor both the numerator and the denominator, if possible.
  • Cancel out common factors between the numerator and the denominator.
  • Rational expressions can often be simplified by substituting and evaluating.

In our example, simplifying wasn't necessary because the given values led to straightforward arithmetic operations. However, knowing how to factor and cancel common factors can make tougher problems easier to solve.
solving step-by-step algebra problems
Solving algebra problems step-by-step helps to break down complex problems into manageable pieces. Follow these methods:

  • Identify the problem and determine what is being asked.
  • Use the substitution method to replace variables with given values.
  • Simplify the expression by performing arithmetic operations.
  • Write down each step to avoid mistakes, ensuring numerical and algebraic steps are correct.

In the exercise: \(\frac{b^{2}+2}{b^{2}-3 b-4}\)
  • For \(b=0\): \(\frac{0^{2}+2}{0^{2}-3 \cdot 0-4} = \frac{2}{-4} = -\frac{1}{2}\)
  • For \(b=2\): \(\frac{(2)^{2}+2}{(2)^{2}-3 \cdot 2-4} = \frac{6}{-6} = -1\)
  • For \(b=-2\): \(\frac{(-2)^{2}+2}{(-2)^{2}-3 \cdot (-2)-4} = \frac{6}{6} = 1\)
Following these steps ensures that the solution is clear and accurate. This approach builds confidence in solving algebraic problems.

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Most popular questions from this chapter

In the following exercises, solve uniform motion applications Jim's speedboat can travel 20 miles upstream against a \(3 \mathrm{mph}\) current in the same amount of time it travels 22 miles downstream with a 3 mph current speed. Find the speed of the Jim's boat.

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Write an inverse variation equation to solve the following problems. Boyle's Law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 psi. (a) Write the equation that relates pressure to volume. (b) If the pressure increases to 330 psi, how much air can Braydon's tank hold?

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