Cumulative Review Exercises Chapters P–2

Simplify.
1. ${(\frac{x^{3}}{y^{2}})}^{2} {(\frac{y^{2}}{x^{3}})}^{3}$ ${(\frac{x^{3}}{y^{2}})}^{2} {(\frac{y^{2}}{x^{3}})}^{3}$
2. $\frac{x^{- 1} y^{- 1}}{x^{- 1} + y^{- 1}}$ $\frac{x^{- 1} y^{- 1}}{x^{- 1} + y^{- 1}}$
Factor.
1. $2 x^{2} + x - 15$ $2 x^{2} + x - 15$
2. $x^{3} - 2 x^{2} + 4 x - 8$ $x^{3} - 2 x^{2} + 4 x - 8$
Combine and simplify.
1. $\sqrt{75} + \sqrt{108} - \sqrt{192}$ $\sqrt{75} + \sqrt{108} - \sqrt{192}$
2. $\frac{x - 1}{x + 1} - \frac{x - 2}{x + 2}$ $\frac{x - 1}{x + 1} - \frac{x - 2}{x + 2}$
Rationalize the denominator.
1. $\frac{1}{2 + \sqrt{3}}$ $\frac{1}{2 + \sqrt{3}}$
2. $\frac{1}{\sqrt{5} - 2}$ $\frac{1}{\sqrt{5} - 2}$

In Exercises 5–9, solve each equation for x.

1. $3 x - 7 = 5$ $3 x - 7 = 5$
2. $\frac{1}{x - 1} = \frac{3}{x - 1}$ $\frac{1}{x - 1} = \frac{3}{x - 1}$
1. $x^{2} - 3 x = 0$ $x^{2} - 3 x = 0$
2. $x^{2} + 3 x - 10 = 0$ $x^{2} + 3 x - 10 = 0$
1. $2 x^{2} - x + 3 = 0$ $2 x^{2} - x + 3 = 0$
2. $4 x^{2} - 12 x + 9 = 0$ $4 x^{2} - 12 x + 9 = 0$
1. $x - 6 \sqrt{x} + 8 = 0$ $x - 6 \sqrt{x} + 8 = 0$
2. ${(x - \frac{1}{x})}^{2} - 10 (x - \frac{1}{x}) + 21 = 0$ ${(x - \frac{1}{x})}^{2} - 10 (x - \frac{1}{x}) + 21 = 0$
1. $\sqrt{3 x - 1} = 2 x - 1$ $\sqrt{3 x - 1} = 2 x - 1$
2. $\sqrt{1 - x} = 2 - \sqrt{2 x + 1}$ $\sqrt{1 - x} = 2 - \sqrt{2 x + 1}$

In Exercises 10–12, solve each inequality for x and write the solution in interval notation.

1. $2 x - 5 < 11$ $2 x - 5 < 11$
2. $- 3 x + 4 > - 5$ $- 3 x + 4 > - 5$
1. $- 3 < 2 x - 3 < 5$ $- 3 < 2 x - 3 < 5$
2. $5 \leq 1 - 2 x \leq 7$ $5 \leq 1 - 2 x \leq 7$
1. $| 2 x - 1 | \leq 7$ $| 2 x - 1 | \leq 7$
2. $| 2 x - 3 | \geq 5$ $| 2 x - 3 | \geq 5$
Show that the triangle with vertices $A (5, - 2), B (6, 5),$ $A (5, - 2), B (6, 5),$ and C(2, 2) is isosceles.
Sketch the graph of $4 x^{2} - 9 y^{2} = 0 .$ $4 x^{2} - 9 y^{2} = 0 .$ [Hint: $4 x^{2} - 9 y^{2} = (2 x + 3 y) (2 x - 3 y) .$ $4 x^{2} - 9 y^{2} = (2 x + 3 y) (2 x - 3 y) .$ ]
Show that the points $(- 3, - 1), (2, 4), (5, 3),$ $(- 3, - 1), (2, 4), (5, 3),$ and (6, 2) lie on a circle with center $(2, - 1) .$ $(2, - 1) .$
Find the center and radius of the circle with equation $x^{2} + y^{2} - 6 x + 4 y + 9 = 0 .$ $x^{2} + y^{2} - 6 x + 4 y + 9 = 0 .$

In Exercises 17–22, find the slope–intercept form of the equation of the line satisfying the given condition.

$Slope = - 3;$ $Slope = - 3;$ y-intercept 5
$Slope = 2;$ $Slope = 2;$ x-intercept 4
The line is perpendicular to $y = 2 x + 3$ $y = 2 x + 3$ and passes through $(2, - 1) .$ $(2, - 1) .$
The line is parallel to $y = 2 x + 3$ $y = 2 x + 3$ and passes through $(2, - 1) .$ $(2, - 1) .$
The line is the perpendicular bisector of the line segment joining $(3, - 1)$ $(3, - 1)$ and (5, 7).
The line is parallel to $x = - 2$ $x = - 2$ and passes through (5, 7).
Find x, assuming that the line through (x, 5) and (5, 11) is parallel to a line with slope 2.
Find x, assuming that the line through (x, 3) and (3, 7) is perpendicular to a line with slope $\frac{1}{2} .$ $\frac{1}{2} .$

In Exercises 25–28, graph each equation.

$12 x = 4 y - 6$ $12 x = 4 y - 6$
$x^{2} - 2 x + y^{2} - 4 y - 4 = 0$ $x^{2} - 2 x + y^{2} - 4 y - 4 = 0$
$f (x) = \sqrt{x + 2} + 3$ $f (x) = \sqrt{x + 2} + 3$
$f (x) = - {(x - 1)}^{2} + 4$ $f (x) = - {(x - 1)}^{2} + 4$
Ms. Gutiérrez bought some used books for $1650. She kept 16 of the books and sold the rest at a profit of $10 each. If she recovered her original $1650 from this sale, how many books did she purchase initially?
The monthly note on a car that was leased for two years was $250 less than the monthly note on a car that was leased for a year and a half. The total expense for the two leases was $21,000. Find the monthly note for each lease.
Let $f (x) = \sqrt{x + 1} - 3$ $f (x) = \sqrt{x + 1} - 3$ .
1. Find the domain of f.
2. Find the intercepts.
3. Find $f (- 1) .$ $f (- 1) .$
4. Solve $f (x) > 0 .$ $f (x) > 0 .$
Let $f (x) = {\begin{array}{l} - x & if x < 0 \\ x^{2} & if x \geq 0 \end{array} .$ $f (x) = {\begin{array}{l} - x & if x < 0 \\ x^{2} & if x \geq 0 \end{array} .$
1. Find $f (- 2), f (0),$ $f (- 2), f (0),$ and f(2).
2. Find the intervals on which f is increasing, decreasing, or constant.
Let $f (x) = \frac{1}{x - 2}$ $f (x) = \frac{1}{x - 2}$ and $g (x) = \frac{2}{x} .$ $g (x) = \frac{2}{x} .$
1. Find $(f \circ g) (x)$ $(f \circ g) (x)$ and state its domain.
2. Find $(g \circ f) (x)$ $(g \circ f) (x)$ and state its domain.

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Table of Contents for Cumulative Review Exercises Chapters P–2