Topic 3: Linear equations and inequalities

Do the same thing to both sides, one line at a time. Reverse the inequality sign when you multiply or divide by a negative.

A lesson for this topic

Part I style: multiple choice

Question 1Part I style

What is the solution of

4(x - 3) + 2x

= 21

Question 2Part I style

What value of

x

makes

7x - 5

= 3x + 19

true?

Question 3Part I style

Solve for

x

\frac{2}{3}x + 4

= 10

Question 4Part I style

Which inequality is the solution of

5 - 3x

> 17

Question 5Part I style

A fitness center charges a one-time fee of $40 plus $25 per month. Dario has $290 to spend on his membership. What is the greatest number of whole months of membership he can pay for?

Question 6Part I style

A school club sells candles for $6 each. The club paid $90 for supplies and wants to make a profit of at least $450. Which inequality can be used to find

c

, the number of candles the club must sell?

Question 7Part I style

k

is a nonzero constant, what is the solution of

kx + 5

= 2kx - 7

for

x

Question 8Part I style

When solving

3(x + 4)

= 21

, Maya writes

3x + 12

= 21

as her first step. Which property justifies this step?

Question 9Part I style

How many solutions does the equation

2(3x - 4)

= 6x - 8

have?

Question 10Part I style

Which equation has no solution?

Question 11Part I style

Noor solved

-2(x - 5)

= 14

as shown. Step 1:

-2x - 10

= 14

Step 2:

-2x

= 24

Step 3:

x

= -12

Which statement about Noor's work is true?

Question 12Part I style

The sum of three consecutive odd integers is 87. If

n

is the smallest of the three integers, which equation can be used to find

n

Question 13Part I style

The length of a rectangular patio is 3 feet more than twice its width. The perimeter of the patio is 69 feet. What is the width of the patio, in feet?

Question 14Part I style

The perimeter of a rectangle is

P

= 2l + 2w

. Which equation gives

w

in terms of

P

and

l

Question 15Part I style

The area of a trapezoid is

A

= \frac{1}{2}h(a + b)

. Which equation gives

b

in terms of

A

h

and

a

Question 16Part I style

The formula

F

= \frac{9}{5}C + 32

converts a temperature in degrees Celsius,

C

, to degrees Fahrenheit,

F

. Which formula gives

C

in terms of

F

Question 17Part I style

ax + by

= c

and

b

\neq 0

, which equation gives

y

in terms of

a

b

c

and

x

Question 18Part I style

A print shop charges $15 to set up a job plus $0.40 for each flyer. The cost, in dollars, of

m

flyers is

C

= 15 + 0.4m

. Which equation gives the number of flyers that can be printed for

C

dollars?

Question 19Part I style

ax + 3

= bx

and

a

\neq b

, what is

x

in terms of

a

and

b

Question 20Part I style

What is the slope of the line that passes through the points

(-2, 7)

and

(4, -5)

Question 21Part I style

Which point lies on the graph of

3x - 2y

= 12

Question 22Part I style

A taxi ride costs $3.50 to start plus $2.25 for each mile driven. Which equation gives the cost,

C

, in dollars, of a ride of

m

miles?

Question 23Part I style

The amount of water,

W

, in gallons, in a tank

t

minutes after it starts to drain is modeled by

W

= 1200 - 35t

. What does the 35 represent?

Question 24Part I style

The graph shows a line that passes through the points

(0, -1)

and

(3, 1)

. Which equation represents the line?

Question 25Part I style

At what value of

x

does the graph of

3x - 4y

= 18

cross the

x

-axis?

Question 26Part I style

Weeks 0, 2, 4 and 6; heights 5, 8, 11 and 14 inches.
Weeks, $x$	Height, $y$ (inches)
0	5
2	8
4	11
6	14

The table shows the height,

y

, in inches, of a bean plant

x

weeks after Lena started measuring it. The heights follow a linear pattern. Which equation models the height of the plant?

Constructed response: Parts II, III and IV

An inequality with parenthesesPart II style · 2 credits

Solve the inequality

8 - 3(x + 2)

\geq 2x - 13

algebraically for

x

(a) Solve

8 - 3(x + 2)

\geq 2x - 13

for

x

, showing your work. [2]

Show your work on paper; type the final answer here. Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

Renting a kayakPart II style · 2 credits

A lakeside shop rents kayaks. The total cost,

T

, in dollars, of renting a kayak for

h

hours is

T

= 12 + 8h

(a) Solve the formula

T

= 12 + 8h

for

h

. [1]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

(b) Javier paid $60 for his rental. For how many hours did he rent the kayak? [1]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

A burning candlePart III style · 4 credits

A candle is 15 inches tall when it is lit. It burns down at a constant rate of 1.25 inches each hour. Let

h

be the height of the candle, in inches,

t

hours after it is lit.

(a) Write an equation for

h

in terms of

t

. [1]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

(b) Complete the table of heights. [1]

Hours, $t$	Height, $h$ (inches)
0	15
2
4
6
8

Type one value in each empty cell.

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

(d) The graph of the equation crosses the $t$ -axis at $t$ $= 12$ . What does this point represent? [1]

A T-shirt fundraiserPart IV style · 6 credits

The science club is selling T-shirts. The club pays a one-time design fee of $180 and $6 for each shirt it orders. It sells every shirt for $15. Let

n

be the number of shirts the club orders and sells.

(a) Write an equation for the club's profit,

P

, in dollars, in terms of

n

. [1]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

(b) The club wants a profit of at least $900. Write an inequality in terms of

n

for this goal. [1]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

n

, showing your work. [2]

Show your work on paper; type the final answer here. Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

(d) Find the club's profit, in dollars, if it sells 150 shirts. [1]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), inequalities as x>=-2 and several solutions as x=2, x=-5.

(e) Maria says that selling 119 shirts is enough to reach the goal. Explain whether she is correct. [1]

0 words · a sentence