College Algebra Review #1 – Step-by-Step Study Guidance

Q1. Consider the functions $f(x) = \frac{1}{2}x - 1$ and $g(x) = x^2 - 4$.

Background

Topic: Functions, Graphing, Intercepts

This question tests your understanding of evaluating functions, graphing them by making a table of values, and finding intercepts.

Key Terms and Formulas:

• x-intercept: The point(s) where the graph crosses the x-axis ($y=0$).

• y-intercept: The point where the graph crosses the y-axis ($x=0$).

• Function evaluation: Substitute the given $x$ value into the function to find $f(x)$ or $g(x)$.

Step-by-Step Guidance

1. For each function, create a table with the given $x$ values: $-2, -1, 0, 1, 2$.

2. For $f(x) = \frac{1}{2}x - 1$, substitute each $x$ value into the formula and simplify to find $f(x)$.

3. For $g(x) = x^2 - 4$, substitute each $x$ value into the formula and simplify to find $g(x)$.

4. To find the $x$-intercepts of $g(x)$, set $g(x) = 0$ and solve for $x$:

   $x^2 - 4 = 0$

5. To find the $y$-intercept of $f(x)$, substitute $x = 0$ into $f(x)$:

   $f(0) = \frac{1}{2}(0) - 1$

Try solving on your own before revealing the answer!

Q2. Solve each equation. If the equation cannot be solved, state that there is no solution.

Background

Topic: Linear Equations

This question tests your ability to solve linear equations, including those with distribution and combining like terms.

Key Terms and Formulas:

• Distributive Property: $a(b + c) = ab + ac$

• Combining Like Terms: Add or subtract terms with the same variable and exponent.

• Solving for $x$: Isolate $x$ on one side of the equation.

Step-by-Step Guidance

1. For each equation, first apply the distributive property where needed.

2. Combine like terms on each side of the equation.

3. Move all terms containing $x$ to one side and constants to the other.

4. Simplify and solve for $x$ by isolating it.

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Q3. Consider the equation $\frac{4}{x+2} + \frac{3}{x} = \frac{10}{x^2 + 2x}$

Background

Topic: Rational Equations

This question tests your ability to solve equations involving rational expressions and to determine restrictions on the variable.

Key Terms and Formulas:

• Rational Expression: A fraction with polynomials in the numerator and/or denominator.

• Restrictions: Values of $x$ that make any denominator zero are not allowed.

• Least Common Denominator (LCD): The smallest expression that all denominators divide into.

Step-by-Step Guidance

1. Identify the denominators: $x+2$, $x$, and $x^2 + 2x$.

2. Set restrictions: $x \neq 0$, $x \neq -2$ (since these make denominators zero).

3. Find the LCD, which is $x(x+2)$.

4. Multiply both sides of the equation by the LCD to clear denominators.

5. Simplify the resulting equation and solve for $x$.

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Q4. Consider the equation $\frac{2x}{x^2 + 6x + 8} + \frac{2}{x+2} = \frac{x}{x+4}$

Background

Topic: Rational Equations

This question tests your ability to solve rational equations and determine restrictions on the variable.

Key Terms and Formulas:

• Factoring: $x^2 + 6x + 8 = (x+2)(x+4)$

• Restrictions: Set denominators not equal to zero.

• LCD: The least common denominator for all terms.

Step-by-Step Guidance

1. Factor $x^2 + 6x + 8$ as $(x+2)(x+4)$.

2. List restrictions: $x \neq -2$, $x \neq -4$.

3. Find the LCD: $(x+2)(x+4)$.

4. Multiply both sides by the LCD to clear denominators.

5. Simplify and solve for $x$.

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Q5. Solve the quadratic equation by the specified method. Report all solutions, real and complex.

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations using different methods: quadratic formula, completing the square, square root property, and factoring.

Key Terms and Formulas:

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Completing the Square: Rearranging the equation to $(x-h)^2 = k$

• Square Root Property: If $x^2 = k$, then $x = \pm \sqrt{k}$

• Factoring: Expressing the quadratic as a product of binomials.

Step-by-Step Guidance

1. For each part, identify the method to use (as specified).

2. Write the equation in standard form $ax^2 + bx + c = 0$ if needed.

3. Apply the appropriate method step by step (set up the quadratic formula, complete the square, etc.).

4. Show the setup for the solution, but stop before the final calculation.

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Q6. An architect is allowed 15 square yards of floor space to add a small bedroom to a house. The width of the rectangular floor must be 7 yards less than two times the length. Find the length and width of the rectangular floor.

Background

Topic: Quadratic Word Problems

This question tests your ability to set up and solve a quadratic equation from a word problem involving area.

Key Terms and Formulas:

• Area of a rectangle: $A = \text{length} \times \text{width}$

• Express width in terms of length: $w = 2l - 7$

Step-by-Step Guidance

1. Let $l$ be the length of the room in yards.

2. Express the width as $w = 2l - 7$.

3. Set up the area equation: $l \times w = 15$.

4. Substitute $w$ into the area equation: $l(2l - 7) = 15$.

5. Expand and rearrange to form a quadratic equation: $2l^2 - 7l - 15 = 0$.

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Q7. Find all real solutions of the following equations. Be sure to check for extraneous solutions when appropriate.

Background

Topic: Solving Polynomial, Radical, and Absolute Value Equations

This question tests your ability to solve higher-degree polynomial equations, radical equations, and absolute value equations, and to check for extraneous solutions.

Key Terms and Formulas:

• Factoring: For polynomials, look for common factors or use grouping.

• Isolate the radical or absolute value before squaring or splitting into cases.

• Check for extraneous solutions: Substitute solutions back into the original equation.

Step-by-Step Guidance

1. For each equation, identify the type (polynomial, radical, absolute value).

2. For polynomials, try factoring or use the Rational Root Theorem.

3. For radical equations, isolate the radical and square both sides.

4. For absolute value equations, set up two cases: one for the positive and one for the negative.

5. Check all solutions in the original equation to rule out extraneous solutions.

Try solving on your own before revealing the answer!

Q8. Solve the compound inequalities. Report solutions in interval notation.

Background

Topic: Compound Inequalities

This question tests your ability to solve and express solutions to compound inequalities.

Key Terms and Formulas:

• Compound Inequality: An inequality with two comparisons, often joined by 'and' or 'or'.

• Interval Notation: Expressing the solution set as an interval.

Step-by-Step Guidance

1. For each compound inequality, isolate $x$ in the middle by performing the same operation on all parts.

2. For 'or' inequalities, solve each part separately and combine the solution sets.

3. Express the final solution in interval notation.

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Q9. Solve the inequalities. Report your answers in interval notation. Start by rewriting your inequality as a compound inequality (without the absolute value bars).

Background

Topic: Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values and to express the solution in interval notation.

Key Terms and Formulas:

• Absolute Value Inequality: $|A| < B$ becomes $-B < A < B$; $|A| \geq B$ becomes $A \leq -B$ or $A \geq B$.

Step-by-Step Guidance

1. Rewrite the absolute value inequality as a compound inequality.

2. Solve for $x$ in each part of the compound inequality.

3. Express the solution in interval notation.

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Q10. Evaluate the function $f(x) = x^2 - 3x + 12$ at the following values or expressions.

Background

Topic: Function Evaluation

This question tests your ability to substitute values and expressions into a function.

Key Terms and Formulas:

• Function Evaluation: Substitute the given value or expression for $x$ in $f(x)$.

Step-by-Step Guidance

1. For $f(-5)$, substitute $x = -5$ into the function and simplify.

2. For $f(2x)$, substitute $x = 2x$ into the function and simplify the expression.

Try solving on your own before revealing the answer!