Solve the following rational equations. a) \frac{1}{3a + 18} = 1 - \frac{8}{a + 6} b)...

Question:

Solve the following rational equations.

a) {eq}\displaystyle\;\frac{1}{3a + 18} = 1 - \frac{8}{a + 6} {/eq}

b) {eq}\displaystyle\;\frac{7}{2x^2 - x - 1} = \frac{5}{x - 1} {/eq}

Rational Equations:

There are two parts to this question. In one rational equation, there are linear terms and in the other, we have quadratic terms. We will apply the splitting the middle term technique to simplify the expression and solve the equation.

Answer and Explanation:

a)

$$\displaystyle\;\frac{1}{3a + 18} = 1 - \frac{8}{a + 6}\\ \frac{1}{3(a+6)}=1-\frac{8}{a+6}\\ \frac{1}{3(a+6)}+\frac{8}{a+6}=1 $$

We will take the L.C.M.

$$\frac{25}{3(a+6)}=1\\ a+6=\frac{25}{3}\\ a=\frac{7}{3} $$

b)

$$\displaystyle\;\frac{7}{2x^2 - x - 1} = \frac{5}{x - 1} $$

We will factorise the quadratic term using the splitting the middle term technique:

$$\frac{7}{2x^2-2x+x-1}=\frac{5}{x-1}\\ \frac{7}{2x(x-1)+1(x-1)}=\frac{5}{x-1}\\ \frac{7}{(x-1)(2x+1)}=\frac{5}{x-1} $$

Cancelling the common term:

$$\frac{7}{2x+1}=5\\ 7=10x+5\\ x=\frac{1}{5} $$


Learn more about this topic:

Loading...
Expressions of Rational Functions

from Precalculus: High School

Chapter 13 / Lesson 4
4.9K

Related to this Question

Explore our homework questions and answers library