College Algebra Midterm Review: Linear Equations, Inequalities, & Functions

Q1. Solve the linear equations:

Background

Topic: Linear Equations

This section tests your ability to solve basic linear equations for the variable x. Linear equations are equations of the first degree, meaning the variable is not raised to any power other than 1.

Key Terms and Formulas:

• Linear Equation: An equation that can be written in the form $ax + b = c$.

• Isolate the variable: Use inverse operations to get x alone on one side of the equation.

Step-by-Step Guidance

1. Distribute any numbers outside parentheses and combine like terms on both sides of the equation.

2. Move all terms containing x to one side and constants to the other side using addition or subtraction.

3. Combine like terms if necessary.

4. Divide both sides by the coefficient of x to solve for x.

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Q2. Solve compound inequalities and graph the solution. Write your answer in interval notation:

Background

Topic: Compound Inequalities

This question tests your ability to solve inequalities involving more than one comparison, such as $a < bx + c \leq d$. You will also need to express your answer in interval notation and graph the solution on a number line.

Key Terms and Formulas:

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

• Interval Notation: A way to describe the set of solutions using parentheses and brackets.

Step-by-Step Guidance

1. Break the compound inequality into two separate inequalities.

2. Solve each inequality for x.

3. Combine the solutions (intersection for 'and', union for 'or').

4. Write the solution in interval notation.

5. Graph the solution on a number line.

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Q3. Solve absolute value inequalities. Graph the solution and write your answer in interval notation:

Background

Topic: Absolute Value Inequalities

This question tests your understanding of how to solve inequalities involving absolute values, and how to represent the solution both graphically and in interval notation.

Key Terms and Formulas:

• Absolute Value Inequality: $|ax + b| < c$ or $|ax + b| \geq c$

• For $|A| < B$, rewrite as $-B < A < B$

• For $|A| \geq B$, rewrite as $A \leq -B$ or $A \geq B$

Step-by-Step Guidance

1. Rewrite the absolute value inequality as two separate inequalities based on the rules above.

2. Solve each inequality for x.

3. Express the solution in interval notation.

4. Graph the solution on a number line.

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Q4. Find the distance between two points:

Background

Topic: Distance Formula

This question tests your ability to use the distance formula to find the distance between two points in the coordinate plane.

Key Formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Step-by-Step Guidance

1. Label the coordinates of the two points as $(x_1, y_1)$ and $(x_2, y_2)$.

2. Subtract the x-coordinates: $x_2 - x_1$.

3. Subtract the y-coordinates: $y_2 - y_1$.

4. Square both differences, add them, and take the square root.

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Q5. Find the slope of the line passing through two points:

Background

Topic: Slope Formula

This question tests your ability to calculate the slope of a line given two points.

Key Formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Step-by-Step Guidance

1. Label the coordinates of the two points as $(x_1, y_1)$ and $(x_2, y_2)$.

2. Subtract the y-coordinates: $y_2 - y_1$.

3. Subtract the x-coordinates: $x_2 - x_1$.

4. Divide the difference in y by the difference in x to find the slope.

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Q6. Graph the equation using the slope and y-intercept: $y = \frac{2}{3}x - 4$

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a line using its slope and y-intercept, and to state the domain and range.

Key Terms and Formulas:

• Slope-Intercept Form: $y = mx + b$

• Slope ($m$): The rate of change of the line.

• Y-intercept ($b$): The point where the line crosses the y-axis.

Step-by-Step Guidance

1. Identify the slope ($m$) and y-intercept ($b$) from the equation.

2. Plot the y-intercept on the graph.

3. From the y-intercept, use the slope to find another point on the line.

4. Draw the line through the points.

5. State the domain and range (for a linear function, both are usually all real numbers).

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Q7. Find the equation of the line passing through two points in slope-intercept form:

Background

Topic: Equation of a Line

This question tests your ability to find the equation of a line given two points, and to write it in the form $y = mx + b$.

Key Terms and Formulas:

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Slope-Intercept Form: $y = mx + b$

Step-by-Step Guidance

1. Find the slope $m$ using the two points.

2. Use one point and the slope in the equation $y = mx + b$ to solve for $b$.

3. Write the final equation in slope-intercept form.

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Q8. Write the equation of the line that passes through a point and is vertical or horizontal; state the slope:

Background

Topic: Special Lines (Vertical and Horizontal)

This question tests your understanding of the equations and slopes of vertical and horizontal lines.

Key Terms and Formulas:

• Vertical Line: $x = a$ (slope is undefined)

• Horizontal Line: $y = b$ (slope is 0)

Step-by-Step Guidance

1. For a vertical line through $(a, b)$, the equation is $x = a$.

2. For a horizontal line through $(a, b)$, the equation is $y = b$.

3. State the slope for each case.

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Q9. Write the equation of the line that passes through a point and is parallel or perpendicular to a given line:

Background

Topic: Parallel and Perpendicular Lines

This question tests your ability to write equations of lines parallel or perpendicular to a given line, passing through a specific point.

Key Terms and Formulas:

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals.

• Use point-slope form: $y - y_1 = m(x - x_1)$

Step-by-Step Guidance

1. Identify the slope of the given line.

2. For parallel lines, use the same slope; for perpendicular, use the negative reciprocal.

3. Plug the slope and the given point into the point-slope form.

4. Simplify to slope-intercept form if needed.

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Q10. Find the domain and range and determine whether the relation is a function:

Background

Topic: Functions, Domain, and Range

This question tests your understanding of the domain (all possible x-values), range (all possible y-values), and the definition of a function (each input has exactly one output).

Key Terms and Formulas:

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

• Function: A relation where each input has exactly one output.

Step-by-Step Guidance

1. List all x-values for the domain and all y-values for the range.

2. Check if any x-value is repeated with a different y-value (if so, it's not a function).

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Q11. Given $f(x) = x^2 - 4x$ and $g(x) = 2x - 5$, find:

Background

Topic: Evaluating Functions

This question tests your ability to evaluate functions for specific values or expressions.

Key Terms and Formulas:

• Function Evaluation: Substitute the given value or expression for x in the function.

Step-by-Step Guidance

1. For $f(-3)$, substitute $x = -3$ into $f(x)$ and simplify.

2. For $g(a+1)$, substitute $x = a+1$ into $g(x)$ and simplify.

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Q12. Refer to the graph of $f$ to answer the following:

Background

Topic: Interpreting Graphs of Functions

This question tests your ability to read and interpret information from the graph of a function, including domain, range, intervals of increase/decrease, and function values.

Key Terms and Formulas:

• Domain: All x-values for which the function is defined.

• Range: All y-values the function attains.

• Increasing/Decreasing Intervals: Where the graph goes up or down as x increases.

• Function Value: The y-value for a given x.

Step-by-Step Guidance

1. Look at the graph to determine the set of x-values (domain) and y-values (range).

2. Identify intervals where the graph rises (increasing) or falls (decreasing).

3. Find the y-values for the given x-values by locating the points on the graph.

4. For $f(x) = 2$, find the x-value(s) where the graph crosses $y = 2$.

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Q13. Given the piecewise function $f(x) = \begin{cases} 3x-2, & x \leq -1 \\ x^2+1, & x > -1 \end{cases}$, find:

Background

Topic: Piecewise Functions

This question tests your ability to evaluate a piecewise function for given values of x.

Key Terms and Formulas:

• Piecewise Function: A function defined by different expressions for different intervals of x.

Step-by-Step Guidance

1. Determine which part of the piecewise function to use based on the value of x.

2. Substitute the value of x into the appropriate expression and simplify.

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Q14. Find the composite functions:

Background

Topic: Composite Functions

This question tests your ability to find and evaluate composite functions, such as $(f \circ g)(x)$, which means $f(g(x))$.

Key Terms and Formulas:

• Composite Function: $(f \circ g)(x) = f(g(x))$

Step-by-Step Guidance

1. Find $g(x)$ or $h(x)$ as needed.

2. Substitute the result into $f(x)$.

3. Simplify the expression.

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Q15. Find the difference quotient for the given function:

Background

Topic: Difference Quotient

This question tests your ability to compute the difference quotient, which is important for understanding rates of change and the foundation of calculus.

Key Formula:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$

Step-by-Step Guidance

1. Find $f(x+h)$ by substituting $x+h$ into the function.

2. Subtract $f(x)$ from $f(x+h)$.

3. Simplify the numerator as much as possible.

4. Divide the result by $h$.

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