# Lesson 21

Rational Equations (Part 2)

• Let’s write and solve some more rational equations.

### Problem 1

Solve $$x-1 = \dfrac{x^2 - 4x + 3}{x+2}$$ for $$x$$.

### Problem 2

Solve $$\frac{4}{4-x} = \frac{5}{4+x}$$ for $$x$$.

### Problem 3

Show that the equation $$\frac{1}{60} = \frac{2x+50}{x(x+50)}$$ is equivalent to $$x^2 - 70x - 3,\!000 = 0$$ for all values of $$x$$ not equal to 0 or -50. Explain each step as you rewrite the original equation.

### Problem 4

Kiran jogs at a speed of 6 miles per hour when there are no hills. He plans to jog up a mountain road, which will cause his speed to decrease by $$r$$ miles per hour. Which expression represents the time, $$t$$, in hours it will take him to jog 8 miles up the mountain road?

A:

$$t=\frac{8-r}{6}$$

B:

$$t=\frac{8}{6+r}$$

C:

$$t=\frac{6+r}{8}$$

D:

$$t=\frac{8}{6-r}$$

### Problem 5

The rational function $$g(x) = \frac{x+10}{x}$$ can be rewritten in the form $$g(x) = c + \frac{r}{x}$$, where $$c$$ and $$r$$ are constants. Which expression is the result?

A:

$$g(x)=x+\frac{10}{x}$$

B:

$$g(x)=1+\frac{10}{x}$$

C:

$$g(x)=x -\frac{10}{x+10}$$

D:

$$g(x)=1-\frac{1}{x+10}$$

(From Unit 2, Lesson 18.)

### Problem 6

For each equation below, find the value(s) of $$x$$ that make it true.

1. $$10 = \frac{1+7x}{7+x}$$
2. $$0.2=\frac{6+2x}{12+x}$$
3. $$0.8= \frac{x}{0.5+x}$$
4. $$3.5=\frac{4+2x}{0.5-x}$$
(From Unit 2, Lesson 20.)

### Problem 7

A softball player has had 8 base hits out of 25 at bats for a current batting average of $$\frac{8}{25}=.320$$.

How many consecutive base hits does she need if she wants to raise her batting average to .400? Explain or show your reasoning.

(From Unit 2, Lesson 20.)