Temple University Math Placement Study Guide

Overview of the Temple University Math Placement Assessment

Assessment Platform: The Math Placement Assessment is administered online through ALEKS, a web-based learning platform.
Assessment Content: ALEKS evaluates a student's current course knowledge through a set of adaptive questions, typically numbering between $20$ and $30$ .
Adaptive Testing: The system chooses questions based on the student's performance on all previous questions. Because of this, every assessment is unique, and it is impossible to predict specific questions in advance.
Available Tools: For certain questions, an on-screen calculator is provided for student use.
Post-Assessment Improvement: After completing the initial assessment, students have access to ALEKS Learning Modules, designed to help improve mathematical skills and prepare for potential reassessment.
Additional Study Resources: * Paul’s Online Notes (College Algebra): http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx * Khan Academy: https://www.khanacademy.org/ * School Yourself: https://schoolyourself.org/

Whole Numbers, Fractions, and Decimals

Problem 1: Computational Arithmetic * Question: Compute $8.3 - 11.2 - 2$ . * Solution: $-4.9$
Problem 2: Fraction Multiplication * Question: Multiply $\frac{5}{2} \times \frac{3}{10}$ . Write the answer as a fraction in simplest form. * Solution: $\frac{3}{4}$
Problem 3: Fraction Division * Question: Divide $\frac{11}{14} \div \frac{5}{6}$ . Write the answer as a fraction in simplest form. * Solution: $\frac{33}{35}$

Percents, Proportions, and Geometry

Problem 4: Fraction to Percentage Conversion * Question: Write $\frac{20}{50}$ as a percentage. * Solution: $40\%$
Problem 5: Calculating Percentages * Question: What is $90\%$ of $89$ ? * Solution: $80.1$
Problem 6: Direct Variation * Question: For a moving object, the force acting on the object varies directly with the object’s acceleration. When a force of $6\,N$ acts on a certain object, the acceleration is $3\,m/s^2$ . If the force is changed to $20\,N$ , find the new acceleration. * Solution: $10\,m/s^2$
Problem 7: Discount Calculations * Question: An item is regularly priced at $\$90$ . It is now discounted by $60\%$ off the regular price. What is the current price? * Solution: $\$36$
Problem 8: Rectangle Perimeter * Question: The perimeter of a rectangle is $166\,\text{units}$ . Find the length of side $WX$ . Write the answer without variables. * Solution: $44\,\text{units}$
Problem 9: Area of Composite Figures * Question: Find the area of the figure where all sides meet at right angles. * Solution: $42\,yd^2$
Problem 10: Circle Circumference and Area * Question: Find the circumference and the area of a circle with a radius of $3\,yd$ . Use $3.14$ for $\pi$ and do not round. * Solution (Circumference): $18.84\,yd$ * Solution (Area): $28.26\,yd^2$
Problem 11: Arc Length * Question: A circle has a radius of $8\,in$ . Find the length $s$ of the arc intercepted by a central angle of $0.9$ radians. Round the final answer to the nearest tenth. * Solution: $7.2\,in$
Problem 12: Joint and Inverse Variation Equations * Question: Write an equation expressing the relationship: $u$ varies jointly with $p$ and $d$ and inversely with $w$ . Use $k$ as the constant of proportionality. * Solution: $u = \frac{kpd}{w}$

Signed Numbers, Linear Equations, and Inequalities

Problem 13: Absolute Value * Question: Evaluate (a) $|14|$ and (b) $|-8|$ . * Solution: (a) $14$ ; (b) $8$
Problem 14: Single Variable Linear Equation * Question: Solve for $u$ : $9.6 = 6u$ . * Solution: $u = 1.6$
Problem 15: Linear Equation with Fraction * Question: Solve for $w$ : $8 = -\frac{4}{w}$ . Simplify fully. * Solution: $w = -\frac{1}{2}$
Problem 16: Literal Equations (Rearranging Formulas) * Question: Solve the equation $A = \frac{1}{2}h(c + d)$ for $d$ . * Solution: $d = \frac{2A}{h} - c$
Problem 17: Multi-Person Financial Word Problem * Question: Kira, Chang, and Henry have a total of $\$70$ . Chang has $3$ times what Henry has. Henry has $\$5$ more than Kira. Find their individual amounts. * Solution: Kira: $\$10$ , Henry: $\$15$ , Chang: $\$45$
Problem 18: Messaging Totals Word Problem * Question: Rita, Frank, and Justin sent a total of $106$ text messages. Justin sent $10$ fewer than Rita. Frank sent $4$ times as many as Justin. Find their individual message counts. * Solution: Rita: $26$ , Frank: $64$ , Justin: $16$
Problem 19: Determining Solution Types * Question (a): $5(2 + w) - w = 10 + 4(w + 1)$ * Solution (a): No solution * Question (b): $2(x - 1) + 8 = 6(2x - 4)$ * Solution (b): $x = 3$
Problem 20: Solving Linear Inequalities * Question: Solve $-29 < -9 + 4v$ for $v$ . * Solution: $v > -5$
Problem 21: Absolute Value Inequalities (Greater Than) * Question: Solve the inequality $|w - 2| \geq 8$ . * Solution: $w \leq -6$ or $w \geq 10$
Problem 22: Number Line Sets and Interval Notation * Question: Graph the set ${x | -7 < x \leq -2}$ and write in interval notation. * Solution: $(-7, -2]$
Problem 23: Absolute Value Inequalities (Less Than) * Question: Graph the solution to $|w + 9| < 2$ on the number line. * Solution: (Range resides between $-11$ and $-7$ )
Problem 24: Quadratic Inequalities * Question: Graph the solution to $(x - 6)(x - 1) < 0$ on the number line. * Solution: (Range resides between $1$ and $6$ )

Lines and Systems of Linear Equations

Problem 25: Point-Slope Equation * Question: A line passes through $(-2, 6)$ and has a slope of $4$ . Write the equation of the line. * Solution: $y - 6 = 4(x + 2)$
Problem 26: Parallel and Perpendicular Slopes (Set 1) * Question: Consider the line $-8x - 4y = -8$ . Find the slope of a line (a) perpendicular and (b) parallel to it. * Solution: (a) $\frac{1}{2}$ ; (b) $-2$
Problem 27: Parallel and Perpendicular Slopes (Set 2) * Question: Consider the line $8x - 7y = 1$ . Find the slope of a line (a) perpendicular and (b) parallel to it. * Solution: (a) $-\frac{7}{8}$ ; (b) $\frac{8}{7}$
Problem 28: Systems of Linear Equations Solutions * Question (a): $-x - 3y = 3$ , $x + 3y = 3$ * Solution (a): No solution * Question (b): $x + 5y = 5$ , $-x - 5y = -5$ * Solution (b): Infinitely many solutions satisfying $y = 1 - \frac{x}{5}$
Problem 29: Airplane Rate Word Problem * Question: An airplane travels $3480\,km$ in $4$ hours against the wind and $3930\,km$ in $3$ hours with the wind. Find the rate of the plane in still air and the wind rate. * Solution: Plane in still air: $1090\,km/hr$ ; Wind: $220\,km/hr$

Relations and Functions

Problem 30: Set Operations * Question: Sets $J = {a, f, j}$ and $L = {a, b, j}$ . Find (a) Union and (b) Intersection. * Solution: (a) $J \cup L = {a, b, f, j}$ ; (b) $J \cap L = {a, j}$
Problem 31: Domain and Range from Graph * Question: Determine domain and range of function $f$ shown on a graph. * Solution: Domain: $(-3, 2]$ ; Range: $(-5, 4]$
Problem 32: Function Transformations * Question: Describe the translation of $y = f(x) - 2$ from $y = f(x)$ . * Solution: Vertical shift downward two units.
Problem 33: Function Composition * Question: If $q(x) = -x - 1$ and $r(x) = 2x^2 + 2$ , find $(q \circ r)(5)$ and $(r \circ q)(5)$ . * Solution: $(q \circ r)(5) = -53$ ; $(r \circ q)(5) = 74$
Problem 34: Function Arithmetic * Question: If $g(x) = x - 1$ and $h(x) = 4x + 4$ , find $(g + h)(x)$ , $(g \cdot h)(x)$ , and $(g - h)(2)$ . * Solution: $(g + h)(x) = 5x + 3$ ; $(g \cdot h)(x) = 4x^2 - 4$ ; $(g - h)(2) = -11$
Problem 35: Inverse Functions and Compositions * Question: Let $g = {(-5, -1), (2, 0), (6, 2), (8, 1)}$ and $h(x) = 3x - 8$ . Find $g^{-1}(2)$ , $h^{-1}(x)$ , and $(h \circ h^{-1})(7)$ . * Solution: $g^{-1}(2) = 6$ ; $h^{-1}(x) = \frac{x + 8}{3}$ ; $(h \circ h^{-1})(7) = 7$

Integer Exponents and Factoring

Problem 36: Binomial Multiplication * Question: Multiply $(x + 2)(x - 5)$ . Simplify the answer. * Solution: $x^2 - 3x - 10$
Problem 37: Factoring Expressions * Question: Factor $12vw^4y^3 - 18v^5w^9$ . * Solution: $6vw^4(2y^3 - 3v^4w^5)$
Problem 38: Least Common Multiple (LCM) of Monomials * Question: Find the LCM of $10w^8v^6u$ and $4u^5w^4v^7$ . * Solution: $20u^5w^8v^7$

Quadratic and Polynomial Functions

Problem 39: Quadratic Equation Roots * Question: Solve for $u$ : $u^2 + u - 6 = 0$ . * Solution: $u = -3, u = 2$
Problem 40: Polynomial Intercepts * Question: Find all x-intercepts and y-intercepts of $f(x) = 2x^3 - 2x^2 - 84x$ . * Solution: X-intercepts: $x = -6, 0, 7$ ; Y-intercept: $y = 0$
Problem 41: Discriminant Analysis * Question: Compute the value of the discriminant and the number of real solutions for $3x^2 - 6x + 2 = 0$ . * Solution: Discriminant is $12$ ; two real solutions.
Problem 42: Graphing Parabola (Basic) * Question: Graph $y = -\frac{x^2}{2}$ using the vertex and two points on each side.
Problem 43: Graphing Parabola (Vertex Form) * Question: Graph $y = (x - 3)^2 + 4$ using the vertex and two points on each side.
Problem 44: Graphing Parabola (Standard Form) * Question: Graph $y = 3x^2 + 24x + 44$ using the vertex and two points on each side.
Problem 45: Complex Quadratic Solving * Question: Solve the equation $(y - 1)^2 = 2y^2 - 12y + 22$ for $y$ . * Solution: $y = 3, 7$
Problem 46: Circle Equations * Question: Graph the circle defined by $(x + 4)^2 + (y - 3)^2 = 25$ .
Problem 47: Identifying Local Maxima * Question (a): Find all values at which $f$ has a local maximum. * Solution (a): $-3, 2$ * Question (b): Find all local maximum values of $f$ . * Solution (b): $3, 4$
Problem 48: Projectile Motion Calculation * Question: A ball thrown upward ( $h_0 = 3\,ft$ , $v_0 = 37\,ft/s$ ) follows $h = 3 + 37t - 16t^2$ . Find when the height is $23\,ft$ . Round to nearest hundredth. * Solution: $t = 0.86\,s$ or $t = 1.45\,s$
Problem 49: Parabola Characteristics * Question: Find x-intercepts and coordinates of the vertex for $y = x^2 - 2x - 35$ . * Solution: X-intercepts: $x = -5, 7$ ; Vertex: $(1, -36)$

Rational Expressions and Functions

Problem 50: Adding Rational Expressions * Question: Add $-\frac{8}{x - 1} + \frac{3 - 2x}{x}$ . Simplify fully. * Solution: $\frac{-2x^2 + 3x + 3}{x^2 - x}$
Problem 51: Subtracting Rational Expressions (Same Denominator) * Question: Subtract $\frac{-5a - 3b}{7a} - \frac{8a + 10b}{7a}$ . Simplify fully. * Solution: $\frac{-13a + 7b}{7a}$
Problem 52: Combining Rational Fractions * Question: Write $\frac{-6a + 5x}{6a} + \frac{9a - 2x}{4a} + 1$ as a single fraction. * Solution: $\frac{27a - 16x}{12a}$
Problem 53: Simplifying Rational Expressions * Question: Simplify $\frac{8v^5y^4}{8v^4 - 12v^3x}$ . * Solution: $\frac{2v^2y^4}{2v - 3x}$
Problem 54: Solving Rational Equations * Question: Solve $\frac{x - 5}{x - 1} + 1 = \frac{x + 6}{x + 4}$ for $x$ . * Solution: $x = -3, 6$
Problem 55: Expressions with Fractional Exponents * Question: Simplify $(y^{-1/3}x^2)^{2/3}$ . Assume variables are positive; use no negative exponents. * Solution: $\frac{x^{4/3}}{y^{2/9}}$
Problem 56: Graphing Rational Functions * Question: Graph the rational function $y = \frac{2}{x - 2}$ .
Problem 57: Multiplying Rational Expressions * Question: Multiply $\frac{x^2 - 2x - 3}{x + 3} \cdot \frac{x - 2}{3x - 9}$ . Simplify fully. * Solution: $\frac{x^2 - x - 2}{3x + 9}$
Problem 58: Simplifying Fractions with Quadratics * Question: Simplify $\frac{5v^2 + 10v - 40}{v^2 - v - 2}$ . * Solution: $\frac{5(v + 4)}{v + 1}$
Problem 59: Restricted Domain of Rational Functions * Question: Find all values of $x$ not in the domain of $h(x) = \frac{x^2 - 8x + 15}{x^2 - 64}$ . * Solution: $x = -8, 8$

Radicals and Rational Exponents

Problem 60: Rationalizing Single Radicals * Question: Rationalize the denominator and simplify $\sqrt{\frac{7}{2}}$ . * Solution: $\frac{\sqrt{14}}{2}$
Problem 61: Simplified Radical Form * Question: Write $\sqrt[3]{16}$ in simplified radical form. * Solution: $2\sqrt[3]{2}$
Problem 62: Negative Fractional Exponents * Question (a): Simplify $4^{-3/2}$ . * Solution (a): $\frac{1}{8}$ * Question (b): Simplify $(\frac{1}{16})^{5/4}$ . * Solution (b): $\frac{1}{32}$
Problem 63: Variable Product with Rational Exponents * Question: Simplify $(b^{1/5}c^4)^{-3/2}$ . Assume positive variables; no negative exponents. * Solution: $\frac{1}{b^{3/10}c^6}$
Problem 64: Simplifying Sums and Products of Radicals * Question (a): $7w\sqrt{50u^3} + u\sqrt{2uw^2}$ * Solution (a): $36\sqrt{2}u^{3/2}w$ * Question (b): $6x\sqrt{45u^5} + u^2\sqrt{80ux^2}$ * Solution (b): $22xu^2\sqrt{5u}$ * Question (c): $y\sqrt{\frac{32}{y v^2}} - 8v\sqrt{18y^3}$ * Solution (c): $-20\sqrt{2}v\sqrt{y^3}$
Problem 65: Graphing Square Root Functions * Question: Graph $f(x) = 3\sqrt{x} + 1$ , including the leftmost point and three others.
Problem 66: Graphing Linear Fractional Functions * Question: Graph the function $f(x) = \frac{-3x + 4}{-x + 3}$ .
Problem 67: Solving Radical Equations * Question: Solve $y - 3 = \sqrt{-3y + 63}$ where $y$ is real. * Solution: $y = 9$
Problem 68: simplifying exponent ratios * Question: Simplify $\frac{z^{1/3}}{z^{1/4}z^{-3/4}}$ . * Solution: $z^{5/6}$
Problem 69: Rationalizing the Denominator with Conjugates * Question: Rationalize $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ . * Solution: $\frac{7 + 2\sqrt{10}}{3}$

Exponentials and Logarithms

Problem 70: Logarithm Expansion * Question: Expand $\log(yx^3)$ so each log has one variable and no exponents. * Solution: $\log y + 3 \log x$
Problem 71: Exponential to Logarithmic Conversion * Question: Rewrite $x = e^9$ as a logarithmic equation. * Solution: $\ln x = 9$
Problem 72: Natural Log Equations * Question: Solve $\ln(x - 7) + \ln 2 = \ln x$ . * Solution: $x = 14$
Problem 73: Solving Natural Logs with Constants * Question: Solve $5 + \ln(x + 4) = 3$ . Round to nearest hundredth. * Solution: $-3.86$
Problem 74: Solving Basic Logarithmic Equations * Question: Solve $\log_7 x = -2$ . * Solution: $\frac{1}{49}$
Problem 75: Log Equations with Arguments * Question: Solve $\log_5(-1 - 4x) = 1$ . * Solution: $x = -\frac{3}{2}$
Problem 76: Variable in Exponent (Powers of 3) * Question: Solve $27^{-x+1} = 81$ . * Solution: $x = -\frac{1}{3}$
Problem 77: Exponential Equations requiring LN * Question: Solve $e^x = 8$ and $7^{-7y} = 9$ . Round to hundredths. * Solution: $x \approx 2.08$ ; $y \approx -0.16$
Problem 78: Coefficient with Natural Log * Question: Solve $4 \ln(x + 2) = 8$ . Round to hundredths. * Solution: $5.39$
Problem 79: Change of Base Formula * Question: Compute $\log_7(6)$ . Round to nearest thousandth. * Solution: $0.921$
Problem 80: Evaluating Logs * Question: Evaluate $\log_2(\frac{1}{16})$ . * Solution: $-4$

Trigonometry

Problem 81: Composition of Trig Functions * Question: Find the exact value of $\sin(\arctan(-\frac{12}{5}))$ . * Solution: $-\frac{12}{13}$
Problem 82: Unit Circle Coordinates * Question: Find the terminal point for $\frac{7\pi}{6}$ radians. * Solution: $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
Problem 83: Coterminal Angles (Small Range) * Question (a): Coterminal with $-315^\circ$ between $0^\circ$ and $360^\circ$ . * Solution (a): $45^\circ$ * Question (b): Coterminal with $\frac{9\pi}{4}$ between $0$ and $2\pi$ . * Solution (b): $\frac{\pi}{4}$
Problem 84: Coterminal Angles (Large Range) * Question (a): Coterminal with $835^\circ$ between $0^\circ$ and $360^\circ$ . * Solution (a): $115^\circ$ * Question (b): Coterminal with $\frac{33\pi}{10}$ between $0$ and $2\pi$ . * Solution (b): $\frac{13\pi}{10}$
Problem 85: Trig Identities Simplification * Question: Simplify $\tan(x) \cos(x)$ . * Solution: $\sin(x)$
Problem 86: Solving Trig Equations (Cosine) * Question: Solve $2 \cos(\theta) + \sqrt{2} = 0$ for $[0, 2\pi)$ . * Solution: $\frac{3\pi}{4}, \frac{5\pi}{4}$
Problem 87: Six Trig Functions from Points * Question: Let $(-3, 7)$ be on the terminal side of $\theta$ . Find $\sin(\theta), \sec(\theta), \tan(\theta)$ . * Solution: $\sin(\theta) = \frac{7}{\sqrt{58}}$ ; $\sec(\theta) = -\frac{\sqrt{58}}{3}$ ; $\tan(\theta) = -\frac{7}{3}$
Problem 88: Radian to Degree Conversion * Question: Convert $\frac{5\pi}{3}$ radians to degrees. * Solution: $300^\circ$
Problem 89: Solving Trig Equations (Sine) * Question: Solve $2 \sin(\theta) - \sqrt{3} = 0$ for $[0, 2\pi)$ . * Solution: $\frac{\pi}{3}, \frac{2\pi}{3}$
Problem 90: Inverse Trig Constants * Question: Find exact value of $\arctan(-1)$ . * Solution: $-\frac{\pi}{4}$
Problem 91: Double-Angle Formulas * Question: Simplify $\cos^2(4\theta) - \sin^2(4\theta)$ . * Solution: $\cos(8\theta)$
Problem 92: Evaluating Reciprocal Trig Functions * Question (a): $\csc(\frac{5\pi}{4})$ * Solution (a): $-\sqrt{2}$ * Question (b): $\cot(\frac{5\pi}{4})$ * Solution (b): $1$
Problem 93: Sum and Difference Formulas * Question: Find the value of $\cos(\frac{3\pi}{7}) \cos(\frac{5\pi}{28}) + \sin(\frac{3\pi}{7}) \sin(\frac{5\pi}{28})$ . * Solution: $\frac{\sqrt{2}}{2}$
Problem 94: Non-Right Triangle Problems * Question: Triangle $ABC$ : $A = 60^\circ$ , $C = 84^\circ$ , $a = 3$ . Find $B$ , $b$ , and $c$ . * Solution: $B = 36^\circ$ , $b = 2.0$ , $c = 3.4$
Problem 95: Reciprocal Multiplication Simplification * Question: Simplify $\cot(x) \csc(x)$ . * Solution: $\cos(x)$
Problem 96: Trig Equation with Square Terms * Question: Solve $-4 \sin(x) = -\cos^2(x) + 4$ for $[0, 2\pi)$ . * Solution: $x = \frac{3\pi}{2}$
Problem 97: Exact Value of Inverse Tangent * Question: Find the value of $\tan^{-1}(-\frac{\sqrt{3}}{3})$ in terms of $\pi$ . * Solution: $-\frac{\pi}{6}$