SAT Exam 1 (Full Math Module 1)

Question 1:
Solve for x: 4x – 7 = 17.

Correct answer: A. 6

Why A is correct:
Add 7 to both sides:
4x = 24
Divide by 4:
x = 6

Why the others are incorrect:
B. 7 is too large.
C. 8 would make 4x – 7 = 25, not 17.
D. 9 would make 4x – 7 = 29.

Question 2:
Solve for x: 5(x – 2) = 10.

Correct answer: B. 4

Why B is correct:
Divide both sides by 5:
x – 2 = 2
Add 2:
x = 4

Why the others are incorrect:
A. 3 gives 5(1) = 5.
C. 5 gives 5(3) = 15.
D. 6 gives 5(4) = 20.

Question 3:
Solve for x: (x / 2) + 3 = 10.

Correct answer: C. 14

Why C is correct:
Subtract 3:
x / 2 = 7
Multiply by 2:
x = 14

Why the others are incorrect:
A. 10 gives 5 + 3 = 8.
B. 12 gives 6 + 3 = 9.
D. 16 gives 8 + 3 = 11.

Question 4:
If 3x + 2y = 16 and y = 2, what is the value of x?

Correct answer: C. 4

Why C is correct:
Substitute y = 2:
3x + 2(2) = 16
3x + 4 = 16
3x = 12
x = 4

Why the others are incorrect:
A. 2 gives 6 + 4 = 10.
B. 3 gives 9 + 4 = 13.
D. 5 gives 15 + 4 = 19.

Question 5:
Solve for x: 3(x + 1) – 4 = 23.

Correct answer: A. 8

Why A is correct:
Distribute:
3x + 3 – 4 = 23
3x – 1 = 23
3x = 24
x = 8

Why the others are incorrect:
B. 10 makes 3(11) – 4 = 29.
C. 12 makes 3(13) – 4 = 35.
D. 14 makes 3(15) – 4 = 41.

Question 6:
Solve for x: |x – 1| = 2, where x > 1.

Correct answer: B. 3

Why B is correct:
Absolute value equations can have two solutions:
x – 1 = 2 or x – 1 = -2
So x = 3 or x = -1
But the condition says x > 1, so only x = 3 works.

Why the others are incorrect:
A. 2 gives |1| = 1.
C. 4 gives |3| = 3.
D. 5 gives |4| = 4.

Question 7:
If 2x + 5 = 3x, what is the value of x?

Correct answer: C. 5

Why C is correct:
Subtract 2x from both sides:
5 = x
So x = 5

Why the others are incorrect:
A. 3 gives 11 ≠ 9.
B. 4 gives 13 ≠ 12.
D. 6 gives 17 ≠ 18.

Question 8:
A gym charges a 10 dollar sign-up fee plus 4 dollars per class. If the total cost is 50 dollars, how many classes were taken?

Correct answer: D. 10

Why D is correct:
Set up the equation:
10 + 4c = 50
4c = 40
c = 10

Why the others are incorrect:
A. 4 classes would cost 26.
B. 6 classes would cost 34.
C. 8 classes would cost 42.

Question 9:
What is the slope of the line through the points (1, 2) and (3, 10)?

Correct answer: A. 4

Why A is correct:
Slope = rise / run = (10 – 2) / (3 – 1) = 8 / 2 = 4

Why the others are incorrect:
B. 3 is too small.
C. 2 is too small.
D. 1 ignores the actual rise and run.

Question 10:
A taxi charges 3 dollars to start a ride and 2 dollars per mile. What is the total cost, in dollars, of a 1.5-mile ride?

Correct answer: B. 6

Why B is correct:
Cost = 3 + 2(1.5) = 3 + 3 = 6

Why the others are incorrect:
A. 5 is too low.
C. 7 is too high.
D. 8 is too high.

Question 11:
The equation y = 3x + 4 models a delivery fee. What is the y-intercept?

Correct answer: C. 4

Why C is correct:
In slope-intercept form y = mx + b, the y-intercept is b.
Here, b = 4.

Why the others are incorrect:
A. 2 is not the constant term.
B. 3 is the slope.
D. 5 is not in the equation.

Question 12:
If f(x) = 3x + 6, what is the value of f(6)?

Correct answer: D. 24

Why D is correct:
Substitute x = 6:
f(6) = 3(6) + 6 = 18 + 6 = 24

Why the others are incorrect:
A. 15 is too low.
B. 18 ignores the +6.
C. 21 is incorrect arithmetic.

Question 13:
If g(x) = 2x – 3, what value of x makes g(x) = 7?

Correct answer: A. 5

Why A is correct:
Set 2x – 3 = 7
2x = 10
x = 5

Why the others are incorrect:
B. 6 gives 9.
C. 7 gives 11.
D. 8 gives 13.

Question 14:
What are the solutions to x² – 4 = 0?

Correct answer: B. x = 2 and x = -2

Why B is correct:
x² – 4 = 0
x² = 4
x = ±2

Why the others are incorrect:
A. ±3 gives x² = 9.
C. ±4 gives x² = 16.
D. ±1 gives x² = 1.

Question 15:
What is the vertex of y = (x + 1)² + 2?

Correct answer: C. (-1, 2)

Why C is correct:
Vertex form is y = (x – h)² + k, so the vertex is (h, k).
Here x + 1 = x – (-1), so h = -1 and k = 2.

Why the others are incorrect:
A, B, and D all have incorrect signs or reversed coordinates.

Question 16:
Which expression is equivalent to (x + 3)(x + 4)?

Correct answer: A. x² + 7x + 12

Why A is correct:
Multiply:
(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12

Why the others are incorrect:
B. Wrong sign on the middle term.
C. Wrong signs on both middle and constant terms.
D. Wrong sign on the constant term.

Question 17:
What is 10% of 180?

Correct answer: A. 18

Why A is correct:
10% = 0.10
0.10 × 180 = 18

Why the others are incorrect:
B. 20 is too high.
C. 22 is too high.
D. 24 is too high.

Question 18:
A jacket originally costs 50 dollars and is on sale for 40 dollars. What is the percent discount?

Correct answer: B. 20%

Why B is correct:
Discount amount = 50 – 40 = 10
Percent discount = 10 / 50 = 0.20 = 20%

Why the others are incorrect:
A. 25% would be a 12.50 discount.
C. 15% would be a 7.50 discount.
D. 10% would be a $5 discount.

Question 19:
If y varies directly with x and y = 21 when x = 3, what is y when x = 6?

Correct answer: C. 42

Why C is correct:
Direct variation means y = kx
21 = 3k, so k = 7
When x = 6, y = 7(6) = 42

Why the others are incorrect:
A. 30 is too low.
B. 36 is too low.
D. 48 is too high.

Question 20:
Solve |x| = 5.

Correct answer: A. x = 5 or x = -5

Why A is correct:
Absolute value means distance from 0, so x can be 5 or -5.

Why the others are incorrect:
B. ±4 gives |x| = 4
C. ±3 gives |x| = 3
D. ±2 gives |x| = 2

Question 21:

Correct answer: C) 7/13

Explanation:

We are given:

x° + y° = 90°

That means x and y are complementary angles. For complementary angles, the following identity is true:

sin(x°) = cos(90° − x°)

Since x + y = 90, we can rewrite x as:

x = 90 − y

Now substitute that into sin x:

sin(x°) = sin(90° − y°)

Use the cofunction identity:

sin(90° − y°) = cos(y°)

And we are told:

cos(y°) = 7/13

So:

sin(x°) = 7/13

Therefore, the correct answer is C) 7/13.

Why the others are wrong:

A) 13/7 is impossible because sine values must stay between -1 and 1.
B) 6/13 would be the other leg ratio in a 5-12-13 or similar setup only if supported, but it is not what the identity gives here.
D) 6/7 is also not supported by the complementary-angle relationship.

The key idea:
When two angles add to 90°, the sine of one equals the cosine of the other. Since cos y = 7/13, it follows immediately that sin x = 7/13.

Question 22:

Correct answer: A) 5/13

Explanation:

We are given that:

sin w° = 5/13

The question asks for:

cos(90° − w°)

Use the cofunction identity:

cos(90° − θ) = sin θ

Let θ = w. Then:

cos(90° − w°) = sin w°

Since sin w° = 5/13, it follows that:

cos(90° − w°) = 5/13

So the correct answer is A) 5/13.

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