ACT Math — Algebra Domain Test

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Question 1

What is the value of \(x\) if \(3x+7=25\)?

A) 4
B) 6
C) 8
D) 10

Show solution

Subtract 7: \(3x=18\).

Divide by 3: \(x=18/3=6\).

Answer: B

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Question 2

Which expression is equivalent to \(4(2x-3)+5\)?

F) \(8x-7\)
G) \(8x-12\)
H) \(6x-7\)
J) \(8x+2\)

Show solution

Distribute and combine constants: \(4(2x-3)+5=8x-12+5=8x-7\).

Answer: F

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Question 3

If \(5x-4=2x+17\), what is \(x\)?

A) 5
B) 6
C) 7
D) 9

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Subtract \(2x\) and add 4: \(3x=21\), so \(x=7\).

Answer: C

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Question 4

For which value of \(x\) is \(2x+5<17\) true?

F) \(x<5\)
G) \(x>6\)
H) \(x>11\)
J) \(x<6\)

Show solution

Subtract 5: \(2x<12\).

Divide by 2: \(x<12/2\), so \(x<6\).

Answer: J

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Question 5

Which expression is equivalent to \(x^2+7x+12\)?

A) \((x+3)(x+4)\)
B) \((x-3)(x-4)\)
C) \((x+2)(x+6)\)
D) \((x-2)(x-6)\)

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The numbers 3 and 4 multiply to 12 and add to 7: \(x^2+7x+12=(x+3)(x+4)\).

Answer: A

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Question 6

The equation \(y=3x-5\) is in slope-intercept form. What is the slope of the line?

F) -5
G) \(-\frac{5}{3}\)
H) \(\frac{1}{3}\)
J) 3

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In \(y=mx+b\), the coefficient \(m\) is the slope.

Here \(m=3\).

Answer: J

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Question 7

A taxi company charges a fixed fee of \(\$4\) plus \(\$2\) per mile. Which equation gives the total cost \(C\), in dollars, for a trip of \(m\) miles?

A) \(C=4m+2\)
B) \(C=6m\)
C) \(C=2m-4\)
D) \(C=2m+4\)

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The variable cost is \(2m\) and the fixed fee is 4: \(C=2m+4\).

Answer: D

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Question 8

The line shown passes through \((-2,-1)\) and \((2,3)\). What is the slope of the line?

F) -1
G) \(\frac{1}{2}\)
H) 1
J) 2

Show solution

Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).

Substitute the two points: \(m=\frac{3-(-1)}{2-(-2)}=\frac44=1\).

Answer: H

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Question 9

If \(3a+2b=18\) and \(a=4\), what is the value of \(b\)?

A) 2
B) 4
C) 3
D) 6

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Substitute \(a=4\): \(12+2b=18\), so \(2b=6\) and \(b=3\).

Answer: C

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Question 10

Which equation is equivalent to \(4x-3y=12\) when solved for \(y\)?

F) \(y=\frac43x-4\)
G) \(y=-\frac43x+4\)
H) \(y=\frac34x-4\)
J) \(y=-\frac34x+4\)

Show solution

\(4x-3y=12\) gives \(-3y=12-4x\): \(y=\frac43x-4\).

Answer: F

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Question 11

A theater sold 120 tickets for a performance. Adult tickets cost \(\$12\) and student tickets cost \(\$8\). Total ticket revenue was \(\$1,200\). How many adult tickets were sold?

A) 40
B) 50
C) 60
D) 80

Show solution

Let \(a\) be adult tickets and \(s\) student tickets: then \(a+s=120\) and \(12a+8s=1200\).

Substitute \(s=120-a\): \(12a+960-8a=1200\). \(4a=240\) and \(a=60\).

Answer: C

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Question 12

What is the solution set of \(|2x-3|=7\)?

F) \(\{-5,2\}\)
G) \(\{2,5\}\)
H) \(\{-5,-2\}\)
J) \(\{-2,5\}\)

Show solution

Solve \(2x-3=7\) or \(2x-3=-7\).

These give \(x=5\) or \(x=-2\).

Answer: J

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Question 13

Which expression is equivalent to \((2x-5)(x+3)\)?

A) \(2x^2+x-15\)
B) \(2x^2-2x-15\)
C) \(2x^2+11x-15\)
D) \(2x^2+x+15\)

Show solution

Multiply: \(2x^2+6x-5x-15=2x^2+x-15\).

Answer: A

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Question 14

The graph shows two lines that intersect at \((2,3)\). Which ordered pair is the solution to the corresponding system of equations?

F) \((-2,3)\)
G) \((2,-3)\)
H) \((2,3)\)
J) \((3,2)\)

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The solution to a graphed system is the intersection point.

The lines intersect at \((2,3)\).

Answer: H

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Question 15

For \(x\ne4\), which expression is equivalent to \(\frac{x^2-16}{x-4}\)?

A) \(x-4\)
B) \(x+4\)
C) \(x^2+4\)
D) \(1\)

Show solution

Factor the numerator: \(x^2-16=(x-4)(x+4)\).

Cancel \(x-4\) to get \(x+4\).

Answer: B

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Question 16

The system \(2x+ky=6\) and \(4x+6y=12\) has infinitely many solutions. What is the value of \(k\)?

F) 2
G) 3
H) 6
J) 12

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For infinitely many solutions, the first equation multiplied by 2 must equal the second.

Thus \(2k=6\), so \(k=3\).

Answer: G

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Question 17

The roots of \(x^2-px+18=0\) are positive integers that differ by 3. What is the value of \(p\)?

A) 11
B) 12
C) 15
D) 9

Show solution

The positive factor pair of 18 that differs by 3 is 3 and 6.

Their sum is \(p\), so \(p=9\).

Answer: D

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Question 18

The graph of \(y=a(x-2)^2-3\) passes through \((0,5)\), as shown. What is the value of \(a\)?

F) 1
G) 2
H) 3
J) 4

Show solution

Substitute \((0,5)\): \(5=a(0-2)^2-3=4a-3\).

Thus \(8=4a\) and \(a=2\).

Answer: G

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Question 19

If \(\frac{3}{x-1}=\frac{2}{x+2}\), what is the value of \(x\)?

A) -8
B) -5
C) 5
D) 8

Show solution

Cross-multiply: \(3(x+2)=2(x-1)\).

Then \(3x+6=2x-2\), so \(x=-8\).

Answer: A

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Question 20

For what value of \(c\) does the equation \(x^2-6x+c=0\) have exactly one real solution?

F) 6
G) 9
H) 12
J) 36

Show solution

Exactly one real solution occurs when the discriminant is 0: \((-6)^2-4(1)c=0\).

Thus \(36-4c=0\), so \(c=9\).

Answer: G

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