This is a comprehensive collection of over 300
problems for MAT 101 – College Algebra,
complete with detailed, stepbystep solutions
for each question. The problems are organized
by topic to cover the standard curriculum.
Topic 1: The Real Number System & Algebraic Expressions
Problem 1.1
Simplify: \( 3(2x 5) 4(x + 1) \)
Solution:
\[
3(2x 5) 4(x + 1) = 6x 15 4x 4 = (6x 4x) + (15 4) = 2x 19
\]
Problem 1.2
Simplify: \( \frac{2}{3} + \frac{3}{4} \frac{1}{6} \)
Solution:
LCD = 12.
\[
\frac{8}{12} + \frac{9}{12} \frac{2}{12} = \frac{15}{12} = \frac{5}{4}
\]
,Problem 1.3
Simplify: \( (2)^3 (3)^2 \)
Solution:
\[
(2)^3 = 8, \quad (3)^2 = 9, \quad 8 9 = 17
\]
Problem 1.4
Simplify: \( \frac{5x^2 y^3}{10 x^4 y} \)
Solution:
\[
\frac{5}{10} \cdot x^{24} \cdot y^{31} = \frac{1}{2} x^{2} y^{2} = \frac{y^2}{2x^2}
\]
Problem 1.5
Simplify: \( \sqrt{50} \sqrt{18} \)
Solution:
\[
\sqrt{50} = 5\sqrt{2}, \quad \sqrt{18} = 3\sqrt{2}, \quad 5\sqrt{2} 3\sqrt{2} = 2\sqrt{2}
,\]
Problem 1.6
Simplify: \( (3x^2 y)(2x y^3) \)
Solution:
\[
3 \cdot (2) \cdot x^{2+1} \cdot y^{1+3} = 6x^3 y^4
\]
Problem 1.7
Simplify: \( \frac{x^2 9}{x^2 6x + 9} \)
Solution:
\[
\frac{(x3)(x+3)}{(x3)^2} = \frac{x+3}{x3}, \quad x \neq 3
\]
Problem 1.8
Simplify: \( \left( \frac{2x^3}{y^2} \right)^4 \)
Solution:
\[
, \frac{2^4 x^{12}}{y^8} = \frac{16x^{12}}{y^8}
\]
Problem 1.9
Simplify: \( \sqrt{75x^3} \)
Solution:
\[
\sqrt{75x^3} = \sqrt{25 \cdot 3 \cdot x^2 \cdot x} = 5x\sqrt{3x}, \quad x \geq 0
\]
Problem 1.10
Simplify: \( (2\sqrt{3} + 3)(2\sqrt{3} 3) \)
Solution:
Difference of squares: \( (2\sqrt{3})^2 3^2 = 4 \cdot 3 9 = 12 9 = 3 \)
Topic 2: Linear Equations and Inequalities
Problem 2.1
Solve: \( 5x 7 = 3x + 9 \)
Solution:
problems for MAT 101 – College Algebra,
complete with detailed, stepbystep solutions
for each question. The problems are organized
by topic to cover the standard curriculum.
Topic 1: The Real Number System & Algebraic Expressions
Problem 1.1
Simplify: \( 3(2x 5) 4(x + 1) \)
Solution:
\[
3(2x 5) 4(x + 1) = 6x 15 4x 4 = (6x 4x) + (15 4) = 2x 19
\]
Problem 1.2
Simplify: \( \frac{2}{3} + \frac{3}{4} \frac{1}{6} \)
Solution:
LCD = 12.
\[
\frac{8}{12} + \frac{9}{12} \frac{2}{12} = \frac{15}{12} = \frac{5}{4}
\]
,Problem 1.3
Simplify: \( (2)^3 (3)^2 \)
Solution:
\[
(2)^3 = 8, \quad (3)^2 = 9, \quad 8 9 = 17
\]
Problem 1.4
Simplify: \( \frac{5x^2 y^3}{10 x^4 y} \)
Solution:
\[
\frac{5}{10} \cdot x^{24} \cdot y^{31} = \frac{1}{2} x^{2} y^{2} = \frac{y^2}{2x^2}
\]
Problem 1.5
Simplify: \( \sqrt{50} \sqrt{18} \)
Solution:
\[
\sqrt{50} = 5\sqrt{2}, \quad \sqrt{18} = 3\sqrt{2}, \quad 5\sqrt{2} 3\sqrt{2} = 2\sqrt{2}
,\]
Problem 1.6
Simplify: \( (3x^2 y)(2x y^3) \)
Solution:
\[
3 \cdot (2) \cdot x^{2+1} \cdot y^{1+3} = 6x^3 y^4
\]
Problem 1.7
Simplify: \( \frac{x^2 9}{x^2 6x + 9} \)
Solution:
\[
\frac{(x3)(x+3)}{(x3)^2} = \frac{x+3}{x3}, \quad x \neq 3
\]
Problem 1.8
Simplify: \( \left( \frac{2x^3}{y^2} \right)^4 \)
Solution:
\[
, \frac{2^4 x^{12}}{y^8} = \frac{16x^{12}}{y^8}
\]
Problem 1.9
Simplify: \( \sqrt{75x^3} \)
Solution:
\[
\sqrt{75x^3} = \sqrt{25 \cdot 3 \cdot x^2 \cdot x} = 5x\sqrt{3x}, \quad x \geq 0
\]
Problem 1.10
Simplify: \( (2\sqrt{3} + 3)(2\sqrt{3} 3) \)
Solution:
Difference of squares: \( (2\sqrt{3})^2 3^2 = 4 \cdot 3 9 = 12 9 = 3 \)
Topic 2: Linear Equations and Inequalities
Problem 2.1
Solve: \( 5x 7 = 3x + 9 \)
Solution:
