Q1. Solve the equation: 6x + 2 = 7(x - 4)

Background

Topic: Solving Linear Equations

This question tests your ability to solve a linear equation in one variable by applying properties of equality and simplifying expressions.

Key Terms and Formulas

• Linear equation: An equation of the form ax + b = cx + d.

• Distributive property: a(b + c) = ab + ac.

• Isolate the variable: Get all terms with x on one side and constants on the other.

Step-by-Step Guidance

1. Apply the distributive property to expand the right side: .

2. Rewrite the equation: .

3. Subtract from both sides to get all x terms on one side: .

4. Simplify the x terms: .

Try solving on your own before revealing the answer!

Q2. Solve the equation: 4x + 7 - 9x - 8 = 4

Background

Topic: Solving Linear Equations

This question tests your ability to combine like terms and solve for x in a linear equation.

Key Terms and Formulas

• Combine like terms: Add or subtract coefficients of the same variable.

• Isolate the variable: Get all x terms on one side and constants on the other.

Step-by-Step Guidance

1. Combine like terms for x: .

2. Combine constants: .

3. Rewrite the equation: .

4. Add 1 to both sides: .

Try solving on your own before revealing the answer!

Q3. Solve the inequality: 7z > 6z - 15

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas

• Inequality: A mathematical sentence that shows the relationship between expressions using >, <, ≤, or ≥.

• Isolate the variable: Get all z terms on one side.

• Interval notation: A way to describe the set of solutions.

Step-by-Step Guidance

1. Subtract from both sides: .

2. Simplify: .

3. Express the solution in interval notation: .

Try solving on your own before revealing the answer!

Q4. Solve the inequality and write the solution set in interval notation: 5(x + 2) ≤ 6(x - 2)

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve inequalities involving distribution and to write the solution in interval notation.

Key Terms and Formulas

• Distributive property: .

• Isolate the variable: Get all x terms on one side.

• Interval notation: Describes the set of solutions.

Step-by-Step Guidance

1. Apply the distributive property: and .

2. Rewrite the inequality: .

3. Subtract from both sides: .

4. Add 12 to both sides: .

Try solving on your own before revealing the answer!

Q5. Graph the function: f(x) = 3

Background

Topic: Graphing Constant Functions

This question tests your understanding of how to graph a constant function on the coordinate plane.

Key Terms and Formulas

• Constant function: , where c is a constant.

• Graph: A horizontal line at .

Step-by-Step Guidance

1. Recognize that means for all x.

2. On the coordinate plane, plot a horizontal line passing through .

3. Label the line appropriately.

Try graphing on your own before revealing the answer!

Q6. Graph the equation: x - 2y = -2

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a linear equation by finding intercepts or rewriting in slope-intercept form.

Key Terms and Formulas

• Slope-intercept form: .

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Step-by-Step Guidance

1. Rewrite the equation in slope-intercept form: becomes .

2. Divide both sides by -2: .

3. Identify the y-intercept (b = 1) and the slope (m = 1/2).

4. Plot the y-intercept at (0, 1), then use the slope to find another point (rise 1, run 2).

Try graphing on your own before revealing the answer!

Q7. Use the vertical line test to determine whether the graph is the graph of a function.

Background

Topic: Functions and Graphs

This question tests your understanding of the vertical line test for functions.

Key Terms and Formulas

• Vertical line test: If any vertical line crosses the graph more than once, it is not a function.

• Function: Each input (x) has exactly one output (y).

Step-by-Step Guidance

1. Examine the graph provided.

2. Imagine drawing vertical lines at various x-values.

3. Check if any vertical line crosses the graph more than once.

Try analyzing before revealing the answer!

Q8. Use the vertical line test to determine whether the graph is the graph of a function.

Background

Topic: Functions and Graphs

This question tests your understanding of the vertical line test for functions.

Key Terms and Formulas

• Vertical line test: If any vertical line crosses the graph more than once, it is not a function.

• Function: Each input (x) has exactly one output (y).

Step-by-Step Guidance

1. Examine the graph provided.

2. Imagine drawing vertical lines at various x-values.

3. Check if any vertical line crosses the graph more than once.

Try analyzing before revealing the answer!

Q9. Find the slope of the line that goes through the points (2, -6) and (8, 1)

Background

Topic: Slope of a Line

This question tests your ability to use the slope formula to find the slope between two points.

Key Terms and Formulas

• Slope formula:

• and are the coordinates of the two points.

Step-by-Step Guidance

1. Label the points: and .

2. Plug the values into the slope formula: .

3. Simplify the numerator and denominator.

Try calculating before revealing the answer!

Q10. Determine whether the lines are parallel, perpendicular, or neither: f(x) = 7x - 13, g(x) = \frac{1}{7}x + 2

Background

Topic: Slopes of Lines

This question tests your understanding of the relationship between slopes of parallel and perpendicular lines.

Key Terms and Formulas

• Parallel lines: Same slope.

• Perpendicular lines: Slopes are negative reciprocals.

• Neither: Slopes are neither equal nor negative reciprocals.

Step-by-Step Guidance

1. Identify the slopes: (from f(x)), (from g(x)).

2. Check if the slopes are equal (parallel) or negative reciprocals (perpendicular).

3. If neither, then the lines are neither parallel nor perpendicular.

Try analyzing before revealing the answer!

Q11. Determine whether the lines are parallel, perpendicular, or neither: f(x) = -8x - 5, g(x) = -8x + 5

Background

Topic: Slopes of Lines

This question tests your understanding of the relationship between slopes of parallel and perpendicular lines.

Key Terms and Formulas

• Parallel lines: Same slope.

• Perpendicular lines: Slopes are negative reciprocals.

• Neither: Slopes are neither equal nor negative reciprocals.

Step-by-Step Guidance

1. Identify the slopes: (from f(x)), (from g(x)).

2. Check if the slopes are equal (parallel).

3. If not, check if they are negative reciprocals (perpendicular).

Try analyzing before revealing the answer!

Q12. Write an equation of the line with slope -3 and passing through (-7, 4). Write in the form y = mx + b.

Background

Topic: Writing Equations of Lines

This question tests your ability to use the point-slope form and convert to slope-intercept form.

Key Terms and Formulas

• Point-slope form:

• Slope-intercept form:

Step-by-Step Guidance

1. Plug the point and slope into the point-slope form: .

2. Distribute the -3: .

3. Add 4 to both sides to solve for y: .

Try writing the final equation before revealing the answer!

Q13. Write an equation of the line with slope 4/7 and passing through (0, 6). Write in the form y = mx + b.

Background

Topic: Writing Equations of Lines

This question tests your ability to use the point-slope form and convert to slope-intercept form.

Key Terms and Formulas

• Point-slope form:

• Slope-intercept form:

Step-by-Step Guidance

1. Plug the point and slope into the point-slope form: .

2. Simplify: .

3. Add 6 to both sides: .

Try writing the final equation before revealing the answer!

Q14. Solve the system of equations: 2x + y = 4, 3x + 2y = 9

Background

Topic: Solving Systems of Linear Equations

This question tests your ability to solve a system using substitution or elimination.

Key Terms and Formulas

• System of equations: Two or more equations with the same variables.

• Substitution method: Solve one equation for one variable and substitute into the other.

• Elimination method: Add or subtract equations to eliminate a variable.

Step-by-Step Guidance

1. Solve the first equation for y: .

2. Substitute into the second equation: .

3. Expand and simplify: .

4. Combine like terms and solve for x.

Try solving before revealing the answer!

Q15. Solve the system of equations: 5x + 4y = 62, 2x + 4y = 68

Background

Topic: Solving Systems of Linear Equations

This question tests your ability to solve a system using elimination or substitution.

Key Terms and Formulas

• System of equations: Two or more equations with the same variables.

• Elimination method: Subtract one equation from the other to eliminate a variable.

Step-by-Step Guidance

1. Subtract the second equation from the first: .

2. Simplify: .

3. Solve for x.

4. Substitute x back into one of the original equations to solve for y.

Try solving before revealing the answer!

Q16. Graph the inequality: x - y > -3

Background

Topic: Graphing Linear Inequalities

This question tests your ability to graph a linear inequality by first graphing the boundary line and then shading the appropriate region.

Key Terms and Formulas

• Boundary line: Replace the inequality with an equation to graph the line.

• Shading: For '>', shade above the line; for '<', shade below.

Step-by-Step Guidance

1. Rewrite the inequality as an equation: .

2. Rewrite in slope-intercept form: .

3. Graph the line (dashed, since the inequality is strict).

4. Test a point (like (0,0)) to determine which side to shade.

Try graphing before revealing the answer!

Q17. Graph the solutions to the system of linear inequalities: y ≥ 2x - 2, y ≤ -4 - x

Background

Topic: Graphing Systems of Linear Inequalities

This question tests your ability to graph two inequalities and find the region where their solutions overlap.

Key Terms and Formulas

• Graph each boundary line: (solid), (solid).

• Shade above for '≥', below for '≤'.

• The solution is the overlapping region.

Step-by-Step Guidance

1. Graph with a solid line and shade above.

2. Graph with a solid line and shade below.

3. The solution set is where the shaded regions overlap.

Try graphing before revealing the answer!