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Solve rational equations

x-1=(2-x)^{2}

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What are the solutions to the given equation?

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1

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x=0

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x=\frac{5-\sqrt{5}}{2}

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x=\frac{5+\sqrt{5}}{2}

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x=\frac{-1-\sqrt{21}}{2}

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x=\frac{-1+\sqrt{21}}{2}

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(D) The equation has no real solutions.

x

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-9 x+5 \leq 17 \quad

\quad 13 x+25 \leq-1

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1

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x \leq-2

x \geq-\frac{4}{3}

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x \leq-2

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x \geq-\frac{4}{3}

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(D) There are no solutions

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(E) All values of

x

x

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-18 x+21>-15 \quad \text { OR } \quad 20 x-13 \geq 27

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1

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x<-2

x \geq 2

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x=-2

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x \geq 2

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D There are no solutions

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(E) All values of

x

x

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5 x-4 \geq 12 \quad

\quad 12 x+5 \leq-4

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1

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x \geq \frac{16}{5}

x \leq-\frac{3}{4}

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x \leq \frac{16}{5}

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x \geq-\frac{3}{4}

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(D) There are no solutions

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(E) All values of

x

x

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12 x+7<-11 \quad

\quad 5 x-8 \geq 40

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1

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x<-\frac{3}{2}

x \geq \frac{48}{5}

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-\frac{3}{2}<x \leq \frac{48}{5}

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x>\frac{3}{2}

x \leq \frac{48}{5}

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(D) There are no solutions

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(E) All values of

x

x

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5 x-4 \geq 12 \quad

\quad 12 x+5 \leq-4

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1

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x \geq \frac{16}{5}

x \leq-\frac{3}{4}

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x \leq \frac{16}{5}

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x \geq-\frac{3}{4}

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(D) There are no solutions

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(E) All values of

x

6 x+2 y=3

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6 x+y=3

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Consider the given system of equations. How many

(x, y)

solutions does this system have?

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1

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(A) No solutions

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(B) Exactly one solution

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(C) Infinitely many solutions

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(D) None of the above

\begin{array}{l} 2 x-1=y \\ 3 x-1=y \end{array}

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Consider the given system of equations. Which of the following statements about this system is true?

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1

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(A) There is only one

(x, y)

y

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(B) There is only one

(x, y)

y

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(C) There are infinitely many

(x, y)

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(D) There are no

(x, y)

s-2=9

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3 r-4 s=16

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Which of the following accurately describes all solutions to the system of equations shown?

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1

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r=8

s=2

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r=20

s=11

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(C) There are infinite solutions to the system.

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D There are no solutions to the system.

\begin{aligned} -6 x+4 y & =2 \\ 3 x-2 y & =-1 \end{aligned}

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Consider the system of equations. How many

(x, y)

solutions does this system have?

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1

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(A) No solutions

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(B) Exactly one solution

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(C) Exactly two solutions

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D Infinitely many solutions

\begin{array}{c} -x+3=y \\ -6 x+18=6 y \end{array}

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Which of the following accurately describes all solutions to the system of equations shown?

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1

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x=0

y=3

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x=3

y=0

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(C) There are infinite solutions to the system.

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D There are no solutions to the system.

\begin{aligned} -20 x+12 y & =24 \\ -5 x+3 y & =6 \end{aligned}

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Consider the system of equations. How many

(x, y)

solutions does this system have?

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1

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(A) No solutions

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(B) Exactly one solution

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(C) Infinitely many solutions

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(D) None of the above

An elevator company installs elevators in new buildings. The number of meters of cable needed for an elevator is

12

times the number of stories in the building plus an extra

15

meters needed to go into the machinery. Which of the following functions best describes how many meters of cable are needed to install the elevator if there are

s

stories in the building?

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1

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f(s)=12 s+15

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f(s)=15 s+12

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f(s)=12 s+12

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f(s)=15 s+15

\begin{aligned} t-5 & =\frac{1}{3}(u+15)-5 \\ t & =\frac{2}{3}(u+9) \end{aligned}

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Which of the following accurately describes all solutions to the system of equations shown?

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1

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u=-18

t=-6

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u=-3

t=4

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(C) There are infinite solutions to the system.

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(D) There are no solutions to the system.

\begin{aligned} w x+2 y & =3(1+y)+1 \\ 8-y & =2(1-y)+3 x \end{aligned}

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In the system of equations,

w

is a constant. For what value of

w

will the system of equations have exactly one solution

(x, y)

x=1

\begin{array}{c} 8(x-9 y)=8-6(x-4) \\ 4 x-A y=16-3 x \end{array}

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In the system of equations,

A

is a constant. For what value of

A

does the system of linear equations have infinitely many solutions?

11 e-9 f+1=5(f-3 e)

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2 f+4 e=3

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Consider the system of equations. Which of the following statements is true?

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1

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(A) There is only one solution

(e, f)

e \cdot f

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(B) There is only one solution

(e, f)

e \cdot f

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(C) There are infinitely many solutions.

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(D) There are no solutions.

0.7 a-0.8 b=-0.1

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a-1.4=-6(b-0.1)

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Consider the system of equations. If

(a, b)

is the solution to the system, then what is the value of the sum of

a

b

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1

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-0

5

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-0

1

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0

1

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0

5

\begin{aligned} 3 x+2 y & =4(x-y-6) \\ 6(y+x) & =7 x-24 \end{aligned}

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Which of the following accurately describes all solutions to the system of equations shown?

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1

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x=3

y=-1

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x=12

y=-2

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(C) There are infinite solutions to the system.

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(D) There are no solutions to the system.

\begin{array}{c} x=\frac{1}{3}(y+4) \\ 4 x-3 y=3(a-x)-2 x \end{array}

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In the system of equations,

a

is a constant. For what value of

a

does the system of linear equations have infinitely many solutions?

4n-30=2(2n+15)

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Which of the following best describes the solutions to the equation shown?

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1

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A) There is exactly one solution,

n=0

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(B) There is exactly one solution,

n=-\frac{15}{4}

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(C) There are no solutions.

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(D) There are infinitely many solutions.