Chapter 1.3, Problem 36AYU

In Problems 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$\frac{2}{x+2}+\frac{3}{x-1}=\frac{-8}{5}$

To determine

To find: The solution for the equation $\frac{1}{x+1} - \frac{5}{x-4} = \frac{21}{4}$ algebraically and verify using a graphing utility.

Answer to Problem 36AYU

Solution:

$x=3$, $\frac{-16}{21}$

Explanation of Solution

Given:

The equation $\frac{1}{x+1} - \frac{5}{x-4} = \frac{21}{4}$

Algebraic solution:

$\frac{1}{x+1} - \frac{5}{x-4} = \frac{21}{4}$

$\frac{1\left(x-4\right)-5\left(x+1\right)}{\left(x+1\right)\left(x-4\right)} = \frac{21}{4}$Take LCM on left hand side

$\frac{x-4-5x-5}{x^2-4x+x-4} = \frac{21}{4}$

$\frac{-4x-9}{x^2-3x-4} = \frac{21}{4}$

$4(-4x-9) = 21(x^2-4x-4)$Cross multiply both sides

$-16x - 36 = 21x^2 - \text{<mn>6</mn><mi>x</mi>} - 84$

$-16x - 36 + 16x = 21x^2 - 4x - 84 + 16x$Add $16x$ to both sides

$-36 + 36 = 21x^2 - 47x - 84 + 36$Add 36 to both sides

$0 = 21x^2 - 47x - 48$

$21x^2 - 47x - 48 = 0$

$21x^2 - 63x + 16x - 48 = 0$Splitting the middle term

$21x\left(x\text{ − 3}\right)\text{ + 16}\left(x\text{ − 3}\right)\text{ = 0}$Taking $21x$ common from first two terms and 16 common from next two terms

$\left(x\text{ − 3}\right)\left(21x\text{ + 16}\right)\text{ = 0}$Taking $x\text{ − 3}$ common factor

$x\text{ − 3 = 0}$or $21x\text{ + 16 = 0}$

$x\text{ − 3 + 3 = 0 + 3}$or $21x\text{ + 16 − 16 = 0 }-16$

$x\text{ = 3}$Or $21x\text{ = −16}$

$x\text{ = 3}$Or $x\text{ = }\frac{-16}{21}$

Check:

Let $x\text{ = 3}$ in the expression in $x$ on the left side of the equation

$\frac{1}{x+1} - \frac{5}{x-4} = \frac{1}{3+1} - \frac{5}{3-4}$

$= \frac{1}{4} -(-5)$

$= \frac{1+20}{4}$Take LCM

$= \frac{21}{4}$

Since the expression on the left side give $\frac{21}{4}$ which is on the right side, the solution $x\text{ = 3}$ checks

Let $x\text{ = }\frac{-16}{21}$ in the expression in $x$ on the left side of the equation

$\frac{1}{x+1}-\frac{5}{x-4}=\frac{1}{\frac{-16}{21}+1}-\frac{5}{\frac{-16}{21}-4}$

$\text{= }\frac{\text{1}}{\frac{\text{-16+21}}{\text{21}}}\text{ - }\frac{\text{5}}{\frac{\text{-16-84}}{\text{21}}}$Take LCM in the denominator

$= \frac{21}{5} + \frac{\left(21\right)\left(5\right)}{100}$

$= \frac{21}{5} + \frac{21}{20}$Take LCM

$= \frac{\left(21\right)\left(4\right)+21}{20}$

$= \frac{84+21}{20}$

$= \frac{105}{20}$Divide the numerator and denominator by 5

$= \frac{21}{4}$Since the expression on the left side gives $\frac{-8}{5}$ which is on the right side, the solution $x = \frac{-1}{8}$ checks.