# Solutions for Lesson 9 Practice Questions

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Contents

## Inequalities

Solve the following inequalities.

a) 3x-5\le 31

3x-5\le 31
3x\le 36
x\le 12

b) -7x > 28

-7x > 28
x< -4

c) 12-3x \ge 12x-18

12-3x\ge 12x-18
12\ge 15x-18
30\ge 15x
2\ge x

d) 9x-10 < -4 + 15x

9x-10 < -4 + 15x
-10 < -4 + 6x
-6 < 6x
-1 < x

## Square Roots

Select all that are correct.

a) \sqrt {(-2)^2} = -2

incorrect

b) \sqrt {(-2)^2} = 2

correct

c) \sqrt {(-2)^2} = -2 or 2

incorrect

d) \sqrt {2^2} = 2

correct

e) \sqrt {2^2} = -2

incorrect

f) \sqrt {2^2} = 2 or -2

incorrect

## Equations with Square Roots

For each of the following, decide if you can solve the equation.  If you can, write the solution in the box, and if not, write impossible.
a) \sqrt {x-3} = 5

\sqrt {x-3} = 5
x-3 = 25
x = 28

b) \sqrt {3x+1} = -4

\sqrt {3x+1} = -4
impossible

c) -\sqrt {5x-1} = 2

-\sqrt {5x-1} = 2
\sqrt {5x-1} = -2
impossible

d) \sqrt {7-2x} = 3

\sqrt {7-2x} = 3
7-2x = 9
-2x = 2
x = -1

e) -\sqrt {1+3x} = -10

-\sqrt {1+3x} = -10
\sqrt {1+3x} = 10
1+3x = 100
3x = 99
x = 33

f) -\sqrt {12-3x} = 0

-\sqrt {12-3x} = 0
\sqrt {12-3x} = 0
12-3x = 0
-3x = -12
x = 4

## Equations with Rational Expressions

Solve the equations.
a) \frac {3}{x} + 5 = 2

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

b) \frac {5}{2x+1} = 2

\frac {5}{2x+1} = 2
5 = 2(2x+1)
5 = 4x + 2
3 = 4x
\frac {3}{4} = x

c) \frac {x^2}{x-1} + 3x = 4x + 2

\frac {x^2}{x-1} + 3x = 4x + 2
\frac {x^2}{x-1} = x + 2
x^2 = (x-1)(x+2)
x^2 = x^2 + x - 2
0=x-2
2 = x

## Solving Equations with Rational Expressions

Solve the equation \frac {3x}{2x-1} - \frac {4}{x} = \frac {3}{2} .

\frac {3x}{2x-1} - \frac {4}{x} = \frac {3}{2}
\frac {3x}{2x-1} \cdot \frac {x}{x} - \frac {4}{x} \cdot \frac {2x-1}{2x-1} = \frac {3}{2}
\frac {3x^2}{2x^2-x} - \frac {8x-4}{2x^2-x} = \frac {3}{2}
\frac {3x^2 - 8x+4}{2x^2-x} = \frac {3}{2}
2(3x^2-8x+4) = 3(2x^2-x)
6x^2-16x+8 = 6x^2-3x
-16x+8=-3x
-13x=-8
x = \frac {8}{13}

## Solving Equations with Rational Expressions 2

Solve the equation \frac {3x+1}{x-1} + \frac {2x-1}{x+1} = 5

\frac {3x+1}{x-1} + \frac {2x-1}{x+1} = 5
\frac {3x+1}{x-1}\cdot \frac {x+1}{x+1}+\frac {2x-1}{x+1}\cdot \frac {x-1}{x-1} = 5
\frac {3x^2+4x+1}{x^2-1} + \frac {2x^2-3x+1}{x^2-1} = 5
\frac {5x^2+x+2}{x^2-1} = 5
5x^2+x+2 = 5(x^2-1)
5x^2+x+2 = 5x^2-5
x+2 = -5
x = -7

## Solving Equations with Rational Expressions 3

Solve the equations.
a) \frac {6}{x} + \frac {4x}{2x-1} = 2

\frac {6}{x} + \frac {4x}{2x-1} = 2
\frac {6}{x}\cdot \frac{2x-1}{2x-1} + \frac {4x}{2x-1}\cdot \frac{x}{x} = 2
\frac {12x-6}{2x^2-x} + \frac {4x^2}{2x^2-x} = 2
\frac {4x^2+12x-6}{2x^2-x} = 2
4x^2+12x-6 = 2(2x^2-x)
4x^2+12x-6=4x^2-2x
12x-6=-2x
14x-6=0
14x=6
x = \frac {6}{14}
x = \frac {3}{7}

b) \frac {4x}{1-x} + \frac {2x}{4-2x} = -5

\frac {4x}{1-x} + \frac {2x}{4-2x} = -5
\frac {4x}{1-x} + \frac {2x}{2(2-x)} = -5
\frac {4x}{1-x} + \frac {x}{2-x} = -5
\frac {4x}{1-x}\cdot \frac {2-x}{2-x} + \frac {x}{2-x}\cdot \frac {1-x}{1-x} = -5
\frac {4x(2-x)}{(1-x)(2-x)} + \frac {x(1-x)}{(1-x)(2-x)} = -5
\frac {8x - 4x^2}{x^2-3x+2} + \frac {x-x^2}{x^2-3x+2} = -5
\frac {-5x^2 + 9x}{x^2-3x+2} = -5
-5x^2+9x = -5(x^2-3x+2)
-5x^2+9x = -5x^2+15x-10
9x=15x-10
-6x=-10
x = \frac {10}{6}
x = \frac {5}{3}

c) \frac {x}{2-x} + \frac {x}{2x-1} = -\frac {1}{2}

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