Lesson 5 Just Act Rational Solidify Understanding

Ready

Use the given triangle to write the values of $\sin A$ , $\cos A$ , $\tan A$ , $\sin B$ , $\cos B$ , and $\tan B$ .

1.

a.

$\sin A$

b.

$\cos A$

c.

$\tan A$

d.

$\sin B$

e.

$\cos B$

f.

$\tan B$

2.

a.

$\sin A$

b.

$\cos A$

c.

$\tan A$

d.

$\sin B$

e.

$\cos B$

f.

$\tan B$

3.

a.

$\sin A$

b.

$\cos A$

c.

$\tan A$

d.

$\sin B$

e.

$\cos B$

f.

$\tan B$

4.

a.

$\sin A$

b.

$\cos A$

c.

$\tan A$

d.

$\sin B$

e.

$\cos B$

f.

$\tan B$

Set

5.

Angela performed the indicated operations on the following rational expressions. Only one of the three problems is correct. Determine which one she answered correctly. Then identify Angela’s errors in the two that are incorrect and correct them.

a.

$\frac{5 x}{(x - 3)} + \frac{2}{(x - 1)}$

$\frac{5 x (x - 1)}{(x - 3) (x - 1)} + \frac{2 (x - 3)}{(x - 3) (x - 1)}$

$\frac{5 x^{2} - x + 2 x - 3}{(x - 3) (x - 1)}$

$\frac{5 x^{2} + x - 3}{(x - 3) (x - 1)}$

Is this problem correct? If not, correct Angela’s work.

b.

$\frac{x}{(x + 3)} - \frac{4 (x + 3)}{(x - 1)}$

$\frac{x}{1} - \frac{4}{(x - 1)}$

$\frac{x (x - 1)}{(x - 1)} - \frac{4}{(x - 1)}$

$\frac{x^{2} - x - 4}{(x - 1)}$

Is this problem correct? If not, correct Angela’s work.

c.

$\frac{(x + 1) (x - 2)}{(x + 2)} \times \frac{(x + 5)}{(x - 2) (x + 2)}$

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

$\frac{(x + 1) (x + 5)}{(x + 2) (x + 2)}$

$\frac{x^{2} + 6 x + 5}{x^{2} + 4 x + 4}$

Is this problem correct? If not, correct Angela’s work.

Rewrite each expression.

6.

$\frac{2 x + 6}{(x + 1)} - \frac{4}{(x + 1)}$

7.

$\frac{2 x}{x + 2} + \frac{x - 1}{x - 5}$

8.

$\frac{x^{2} + 6 x + 8}{x^{2} - 5 x + 4} \cdot \frac{x^{2} + 3 x - 4}{x^{2} + 4 x + 4}$

9.

$\frac{4 x + 8}{5 x - 20} \div \frac{x^{2} - 3 x - 10}{x^{2} - 4 x}$

10.

$\frac{2 x}{(x^{2} - 4)} + \frac{4}{(x + 2)}$

11.

$\frac{x - 10}{x - 4} - \frac{x + 2}{4 - x}$

12.

Rational numbers and rational expressions are comparable because they have similar features. Complete the table by writing the comparable situation for each statement written.

Rational Numbers	Rational Expressions
Whole numbers are rational numbers with a denominator of $1$ .
	Rational expressions are undefined when the denominator is equal to $0$ .
When you add, subtract, multiply, or divide two rational numbers, the result is also a rational number.
Rational fractions are classified as proper fractions when the numeric value of the numerator is smaller than the denominator.

Go

Identify the values of $x$ for which the function is undefined, if any.

13.

$f (x) = \frac{10}{x - 4}$

14.

$f (x) = \frac{22}{x}$

15.

$f (x) = \frac{x - 7}{x + 15}$

16.

$f (x) = \frac{2 x}{5}$

Use $g (x) = \frac{2}{x + 5}$ for problems 17–21.

17.

Describe the end behavior of $g (x)$ .

18.

State the domain and range of $g (x)$ .

19.

Identify the intercepts, if any.

20.

Write the equations of the asymptotes.

Horizontal:

Vertical:

21.

Graph $g (x) = \frac{2}{x + 5}$ . (Include any asymptotes with dotted lines.)