Chapter 8: Q. 319 (page 982)

In the following exercises, solve.
$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16}$

Short Answer

Expert verified

The solution of the equation $\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16}$ is

$x = \frac{- 4}{5}$

Step by step solution

Step 1. Given information

The given equation is

$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16}$

Step 2, Variable values for which denominators are zero

Factor all the denominators of the equation

$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16} \frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{(x + 4) (x - 4)}$

Terms of the equation are not defined for $x = 4 & x = - 4$

so $\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{(x + 4) (x - 4)}, x = 4, x = - 4$

Step 3. Solution of equation

The least common denominator of all denominators in the equation is $(x + 4) (x - 4)$

multiply both sides of the equation by the least common denominator $(x + 4) (x - 4)$

role="math" localid="1647802390820" $(x + 4) (x - 4) (\frac{3}{x + 4} + \frac{7}{x - 4}) = (x + 4) (x - 4) (\frac{8}{x^{2} - 16}) (x + 4) (x - 4) (\frac{3}{x + 4}) + (x + 4) (x - 4) (\frac{7}{x - 4}) = (x + 4) (x - 4) (\frac{8}{x^{2} - 16}) (x - 4) (3) + (x + 4) (7) = 8 3 x - 12 + 7 x + 28 = 8 10 x = - 8 x = \frac{- 4}{5}$

so the solution of the equation isrole="math" localid="1647802536523" $x = \frac{- 4}{5}$

Step 4. Verification

Substitute $x = \frac{- 4}{5}$ in the equation

$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16} \frac{3}{(\frac{- 4}{5}) + 4} + \frac{7}{(\frac{- 4}{5}) - 4} \overset{?}{=} \frac{8}{{(\frac{- 4}{5})}^{2} - 16} \frac{3 \cdot 5}{- 4 + 20} + \frac{7 \cdot 5}{- 4 - 20} \overset{?}{=} \frac{8 \cdot 25}{16 - 16 \cdot 25} \frac{15}{16} - \frac{35}{24} \overset{?}{=} - \frac{25}{48} - \frac{25}{48} = - \frac{25}{48}$

The left-hand side is equal to the right-hand side so the solution is correct.

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In the following exercises, solve.
$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16}$

Short Answer

Step by step solution

Step 1. Given information

Step 2, Variable values for which denominators are zero

Step 3. Solution of equation

Step 4. Verification

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In the following exercises, solve.3x+4+7x-4=8x2-16

Short Answer

Step by step solution

Step 1. Given information

Step 2, Variable values for which denominators are zero

Step 3. Solution of equation

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Decision Maths

Applied Mathematics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In the following exercises, solve.
$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16}$