Problem 1
If \(3(x - 4) + 2 = 5x - 6\), what is the value of \(x\)?
A) \(-1\)
B) \(-2\)
C) \(1\)
D) \(2\)
E) \(4\)
Solution:
Distribute: \(3x - 12 + 2 = 5x - 6\)
Combine: \(3x - 10 = 5x - 6\)
\(-10 + 6 = 5x - 3x \Rightarrow -4 = 2x\)
\(x = \boxed{-2}\)
Problem 2
A line passes through the points \((2, -3)\) and \((6, 5)\). What is the \(y\)-intercept of the line?
A) \(-5\)
B) \(-3\)
C) \(1\)
D) \(-7\)
E) \(2\)
Solution:
Slope: \(m = \dfrac{5 - (-3)}{6 - 2} = \dfrac{8}{4} = 2\)
Use \(y = mx + b\) with point \((2, -3)\): \(-3 = 2(2) + b\)
\(-3 = 4 + b \Rightarrow b = \boxed{-7}\)
Problem 3
Simplify: $$\frac{(2x^3 y^{-2})^2}{4x^{-1} y^3}$$
A) \(\dfrac{x^7}{y^7}\)
B) \(\dfrac{x^5}{y}\)
C) \(x^7 y^7\)
D) \(\dfrac{x^7}{y}\)
E) \(\dfrac{2x^7}{y^7}\)
Solution:
Square the numerator: \((2x^3 y^{-2})^2 = 4x^6 y^{-4}\)
Divide: \(\dfrac{4x^6 y^{-4}}{4x^{-1} y^3} = x^{6-(-1)} \, y^{-4-3} = x^7 y^{-7}\)
\(= \boxed{\dfrac{x^7}{y^7}}\)
Problem 4
The sum of two numbers is \(24\) and their difference is \(6\). What is the product of the two numbers?
A) \(108\)
B) \(120\)
C) \(135\)
D) \(140\)
E) \(144\)
Solution:
Let the numbers be \(a\) and \(b\): \(a + b = 24\), \(a - b = 6\)
Add equations: \(2a = 30 \Rightarrow a = 15\)
Then \(b = 24 - 15 = 9\)
Product: \(15 \times 9 = \boxed{135}\)
Problem 5
Factor completely: \(6x^2 - 7x - 3\).
A) \((6x + 1)(x - 3)\)
B) \((2x + 3)(3x - 1)\)
C) \((3x - 3)(2x + 1)\)
D) \((6x - 3)(x + 1)\)
E) \((2x - 3)(3x + 1)\)
Solution:
Find two numbers multiplying to \(6 \cdot (-3) = -18\) and adding to \(-7\): they are \(-9\) and \(2\).
Split: \(6x^2 - 9x + 2x - 3\)
Group: \(3x(2x - 3) + 1(2x - 3)\)
\(= \boxed{(2x - 3)(3x + 1)}\)
Problem 6
If \(f(x) = 2x^2 - 3x + 1\), what is \(f(-2)\)?
A) \(3\)
B) \(15\)
C) \(9\)
D) \(11\)
E) \(17\)
Solution:
Substitute \(x = -2\): \(2(-2)^2 - 3(-2) + 1\)
\(= 2(4) + 6 + 1\)
\(= 8 + 6 + 1 = \boxed{15}\)
Problem 7
What is the solution set of the inequality \(-3(2x - 4) \geq 18\)?
A) \(x \geq -1\)
B) \(x \leq 1\)
C) \(x \geq 1\)
D) \(x \leq -1\)
E) \(x \leq 5\)
Solution:
Distribute: \(-6x + 12 \geq 18\)
Subtract 12: \(-6x \geq 6\)
Divide by \(-6\) (flip the inequality): \(x \leq -1\)
Answer: \(\boxed{x \leq -1}\)
Problem 8
A car travels \(120\) miles in \(2\) hours and then \(90\) miles in the next \(1.5\) hours. What is the car's average speed for the entire trip, in miles per hour?
A) \(60\)
B) \(55\)
C) \(57\)
D) \(62\)
E) \(65\)
Solution:
Total distance: \(120 + 90 = 210\) miles
Total time: \(2 + 1.5 = 3.5\) hours
Average speed: \(\dfrac{210}{3.5} = \boxed{60}\) mph
Problem 9
For what value of \(k\) does the system below have no solution? $$\begin{cases} 3x - 2y = 7 \\ kx - 4y = 5 \end{cases}$$
A) \(3\)
B) \(4\)
C) \(5\)
D) \(8\)
E) \(6\)
Solution:
No solution means the lines are parallel: coefficients proportional but constants not.
\(\dfrac{k}{3} = \dfrac{-4}{-2} = 2 \Rightarrow k = 6\)
Check not identical: \(\dfrac{5}{7} \neq 2\) ✓ (truly no solution)
Answer: \(k = \boxed{6}\)
Problem 10
If \(x + \dfrac{1}{x} = 5\), what is the value of \(x^2 + \dfrac{1}{x^2}\)?
A) \(21\)
B) \(25\)
C) \(23\)
D) \(27\)
E) \(29\)
Solution:
Square both sides: \(\left(x + \dfrac{1}{x}\right)^2 = 5^2 = 25\)
Expand: \(x^2 + 2 \cdot x \cdot \dfrac{1}{x} + \dfrac{1}{x^2} = 25\)
\(x^2 + 2 + \dfrac{1}{x^2} = 25\)
\(x^2 + \dfrac{1}{x^2} = \boxed{23}\)
Problem 11
Solve for \(x\): \(2x + 5 = 3x - 7\).
A) \(12\)
B) \(-12\)
C) \(2\)
D) \(-2\)
E) \(6\)
Solution:
Move variables to one side: \(5 + 7 = 3x - 2x\)
\(12 = x\)
\(x = \boxed{12}\)
Problem 12
What is the slope of the line \(4x - 2y = 8\)?
A) \(-2\)
B) \(2\)
C) \(4\)
D) \(-4\)
E) \(\tfrac{1}{2}\)
Solution:
Solve for \(y\): \(-2y = -4x + 8\)
\(y = 2x - 4\)
Slope is the coefficient of \(x\): \(\boxed{2}\)
Problem 13
Simplify: \((3x^2)(4x^3)\).
A) \(7x^5\)
B) \(12x^6\)
C) \(12x^5\)
D) \(7x^6\)
E) \(12x^9\)
Solution:
Multiply coefficients: \(3 \cdot 4 = 12\)
Add exponents: \(x^2 \cdot x^3 = x^{2+3} = x^5\)
Result: \(\boxed{12x^5}\)
Problem 14
If \(5^x = 125\), find \(x\).
A) \(5\)
B) \(25\)
C) \(2\)
D) \(3\)
E) \(15\)
Solution:
Write \(125\) as a power of \(5\): \(125 = 5^3\)
So \(5^x = 5^3\)
\(x = \boxed{3}\)
Problem 15
What is the sum of the solutions of \(x^2 - 5x + 6 = 0\)?
A) \(1\)
B) \(6\)
C) \(2\)
D) \(3\)
E) \(5\)
Solution:
Factor: \((x - 2)(x - 3) = 0\)
Solutions: \(x = 2\) and \(x = 3\)
Sum: \(2 + 3 = \boxed{5}\) (or use \(-\tfrac{b}{a} = 5\))
Problem 16
A rectangle's length is \(3\) more than its width. If the perimeter is \(26\), what is the area?
A) \(40\)
B) \(35\)
C) \(30\)
D) \(48\)
E) \(24\)
Solution:
Let width \(= w\), length \(= w + 3\). Perimeter: \(2(w + w + 3) = 26\)
\(4w + 6 = 26 \Rightarrow 4w = 20 \Rightarrow w = 5\)
Length \(= 8\); Area \(= 5 \times 8 = \boxed{40}\)
Problem 17
Simplify: \(\sqrt{48}\).
A) \(2\sqrt{12}\)
B) \(4\sqrt{3}\)
C) \(3\sqrt{4}\)
D) \(16\sqrt{3}\)
E) \(4\sqrt{12}\)
Solution:
Find the largest perfect-square factor: \(48 = 16 \cdot 3\)
\(\sqrt{48} = \sqrt{16}\,\sqrt{3} = 4\sqrt{3}\)
Answer: \(\boxed{4\sqrt{3}}\)
Problem 18
What is \(15\%\) of \(80\)?
A) \(8\)
B) \(16\)
C) \(12\)
D) \(20\)
E) \(10\)
Solution:
Convert percent to decimal: \(15\% = 0.15\)
Multiply: \(0.15 \times 80 = 12\)
Answer: \(\boxed{12}\)
Problem 19
Solve the proportion: \(\dfrac{3}{4} = \dfrac{x}{20}\).
A) \(12\)
B) \(18\)
C) \(20\)
D) \(15\)
E) \(16\)
Solution:
Cross-multiply: \(4x = 3 \cdot 20 = 60\)
\(x = \dfrac{60}{4} = 15\)
Answer: \(\boxed{15}\)
Problem 20
If \(2x - 3y = 12\) and \(x = 6\), find \(y\).
A) \(2\)
B) \(-2\)
C) \(4\)
D) \(6\)
E) \(0\)
Solution:
Substitute \(x = 6\): \(2(6) - 3y = 12\)
\(12 - 3y = 12 \Rightarrow -3y = 0\)
\(y = \boxed{0}\)
Problem 21
Simplify: \(\dfrac{x^2 - 16}{x - 4}\) (for \(x \neq 4\)).
A) \(x + 4\)
B) \(x - 4\)
C) \(x + 16\)
D) \(4\)
E) \(x^2 + 4\)
Solution:
Factor the numerator (difference of squares): \(x^2 - 16 = (x - 4)(x + 4)\)
\(\dfrac{(x - 4)(x + 4)}{x - 4}\)
Cancel \((x - 4)\): \(\boxed{x + 4}\)
Problem 22
An arithmetic sequence has first term \(5\) and common difference \(3\). What is the 10th term?
A) \(30\)
B) \(32\)
C) \(35\)
D) \(27\)
E) \(29\)
Solution:
Use \(a_n = a_1 + (n - 1)d\)
\(a_{10} = 5 + (10 - 1)(3) = 5 + 27\)
\(a_{10} = \boxed{32}\)
Problem 23
What is the sum of all solutions of \(|2x - 1| = 7\)?
A) \(4\)
B) \(-3\)
C) \(1\)
D) \(7\)
E) \(0\)
Solution:
Case 1: \(2x - 1 = 7 \Rightarrow x = 4\)
Case 2: \(2x - 1 = -7 \Rightarrow x = -3\)
Sum: \(4 + (-3) = \boxed{1}\)
Problem 24
Multiply: \((x + 3)(x - 5)\).
A) \(x^2 + 2x - 15\)
B) \(x^2 - 8x - 15\)
C) \(x^2 + 8x + 15\)
D) \(x^2 - 2x - 15\)
E) \(x^2 - 15\)
Solution:
FOIL: \(x \cdot x + x(-5) + 3 \cdot x + 3(-5)\)
\(= x^2 - 5x + 3x - 15\)
\(= \boxed{x^2 - 2x - 15}\)
Problem 25
A shirt costs \(\$40\) after a \(20\%\) discount. What was the original price?
A) \(\$44\)
B) \(\$48\)
C) \(\$60\)
D) \(\$45\)
E) \(\$50\)
Solution:
After 20% off, you pay 80% of the original: \(0.80 \cdot p = 40\)
\(p = \dfrac{40}{0.80} = 50\)
Original price: \(\boxed{\$50}\)
Problem 26
Solve the system for \(x\): \(x + y = 10\), \(x - y = 4\).
A) \(7\)
B) \(3\)
C) \(5\)
D) \(6\)
E) \(14\)
Solution:
Add the two equations: \((x + y) + (x - y) = 10 + 4\)
\(2x = 14\)
\(x = \boxed{7}\)
Problem 27
What is the value of \(x\) in \(3^{2x} = 81\)?
A) \(4\)
B) \(2\)
C) \(3\)
D) \(8\)
E) \(1\)
Solution:
Write \(81\) as a power of \(3\): \(81 = 3^4\)
So \(3^{2x} = 3^4 \Rightarrow 2x = 4\)
\(x = \boxed{2}\)
Problem 28
Simplify: \(\dfrac{6x^4 y^2}{2x y^5}\).
A) \(3x^3 y^3\)
B) \(\dfrac{3x^5}{y^3}\)
C) \(\dfrac{3x^3}{y^3}\)
D) \(\dfrac{4x^3}{y^3}\)
E) \(\dfrac{3x^3}{y^7}\)
Solution:
Coefficients: \(\dfrac{6}{2} = 3\)
Subtract exponents: \(x^{4-1} = x^3\), \(y^{2-5} = y^{-3}\)
\(= \boxed{\dfrac{3x^3}{y^3}}\)
Problem 29
If \(f(x) = x^2 + 2x\), find \(f(3) - f(1)\).
A) \(8\)
B) \(10\)
C) \(15\)
D) \(12\)
E) \(18\)
Solution:
\(f(3) = 3^2 + 2(3) = 9 + 6 = 15\)
\(f(1) = 1^2 + 2(1) = 1 + 2 = 3\)
\(f(3) - f(1) = 15 - 3 = \boxed{12}\)
Problem 30
The sum of three consecutive integers is \(72\). What is the largest of the three?
A) \(23\)
B) \(24\)
C) \(22\)
D) \(26\)
E) \(25\)
Solution:
Let the integers be \(n, n+1, n+2\): \(3n + 3 = 72\)
\(3n = 69 \Rightarrow n = 23\)
Largest \(= n + 2 = \boxed{25}\)