Step-by-Step Guidance for Beginning Algebra Practice Final

Q1. Solve the equation $2(3y - 5) = 3(2y - 3) - 1$ for $y$.

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation with variables on both sides.

Key Terms and Formulas

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

• Combining like terms

• Solving for a variable

Step-by-Step Guidance

1. Apply the distributive property to both sides: $2(3y - 5)$ and $3(2y - 3)$.

2. Expand both sides: $2 \times 3y - 2 \times 5$ and $3 \times 2y - 3 \times 3$.

3. Combine like terms and simplify each side.

4. Move all terms involving $y$ to one side and constants to the other.

Try solving on your own before revealing the answer!

Q2. Solve the equation $\frac{2}{3}(x - 1) = \frac{1}{2}(x + 2)$ for $x$.

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve equations involving fractions.

Key Terms and Formulas

• Multiplying both sides by the least common denominator (LCD) to clear fractions

• Solving for $x$

Step-by-Step Guidance

1. Identify the LCD for the denominators (3 and 2).

2. Multiply both sides of the equation by the LCD to eliminate fractions.

3. Expand and simplify both sides.

4. Collect like terms and solve for $x$.

Try solving on your own before revealing the answer!

Q3. Solve the equation $r = 2st - \frac{1}{3}ar$ for $a$.

Background

Topic: Literal Equations (Solving for a variable)

This question tests your ability to rearrange an equation to solve for a specific variable.

Key Terms and Formulas

• Isolating a variable

• Factoring and combining like terms

Step-by-Step Guidance

1. Move all terms involving $a$ to one side of the equation.

2. Factor $a$ out if necessary.

3. Divide both sides by the coefficient of $a$ to isolate it.

Try solving on your own before revealing the answer!

Q4. Solve the equation $|2x - 1| = 3$ for $x$.

Background

Topic: Absolute Value Equations

This question tests your understanding of how to solve equations involving absolute values.

Key Terms and Formulas

• Absolute value definition: $|A| = B$ means $A = B$ or $A = -B$

Step-by-Step Guidance

1. Set up two equations: $2x - 1 = 3$ and $2x - 1 = -3$.

2. Solve each equation for $x$.

Try solving on your own before revealing the answer!

Q5. Solve the inequality $| -2y + 3| + 2 < 1$ for $y$.

Background

Topic: Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values.

Key Terms and Formulas

• Isolate the absolute value expression before solving

• Absolute value inequalities: $|A| < B$ means $-B < A < B$

Step-by-Step Guidance

1. Subtract 2 from both sides to isolate the absolute value: $| -2y + 3| < -1$.

2. Consider what it means for an absolute value to be less than a negative number.

3. Determine if there are any solutions based on the properties of absolute value.

Try solving on your own before revealing the answer!

Q6. Graph the solution set of the inequality $-3x + 7 > -8$.

Background

Topic: Linear Inequalities

This question tests your ability to solve and graph linear inequalities.

Key Terms and Formulas

• Solving inequalities

• Graphing on a number line

Step-by-Step Guidance

1. Subtract 7 from both sides to isolate the $x$ term.

2. Divide both sides by $-3$ (remember to reverse the inequality sign when dividing by a negative).

3. Express the solution in terms of $x$.

Try solving on your own before revealing the answer!

Q7. Solve the inequality $2(3 - y) + 5y < 3(y - 2) + 2$ for $y$.

Background

Topic: Linear Inequalities

This question tests your ability to solve inequalities with variables on both sides.

Key Terms and Formulas

• Distributive property

• Combining like terms

• Solving inequalities

Step-by-Step Guidance

1. Expand both sides using the distributive property.

2. Combine like terms on each side.

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

4. Solve for $y$.

Try solving on your own before revealing the answer!

Q8. Solve the compound inequality $2x + 3 \leq -1$ or $-2x - 3 \leq -5$ for $x$.

Background

Topic: Compound Inequalities

This question tests your ability to solve compound inequalities and interpret the solution set.

Key Terms and Formulas

• Solving inequalities

• "Or" means the solution is any value that satisfies either inequality

Step-by-Step Guidance

1. Solve $2x + 3 \leq -1$ for $x$.

2. Solve $-2x - 3 \leq -5$ for $x$.

3. Combine the solution sets using "or".

Try solving on your own before revealing the answer!

Q9. Solve the compound inequality $-3 \leq -\frac{1}{3}x + 1 < 2$ for $x$.

Background

Topic: Compound Inequalities

This question tests your ability to solve compound inequalities involving fractions.

Key Terms and Formulas

• Solving compound inequalities

• Multiplying or dividing by negatives reverses the inequality sign

Step-by-Step Guidance

1. Subtract 1 from all parts of the inequality.

2. Multiply all parts by $-3$ to clear the fraction and reverse the inequality signs.

3. Solve for $x$ in the resulting inequality.

Try solving on your own before revealing the answer!

Q10. Find the $x$-intercept of the graph of $x - 3y = 6$.

Background

Topic: Linear Equations and Graphs

This question tests your ability to find the $x$-intercept of a line.

Key Terms and Formulas

• $x$-intercept: The point where $y = 0$

Step-by-Step Guidance

1. Set $y = 0$ in the equation $x - 3y = 6$.

2. Solve for $x$.

Try solving on your own before revealing the answer!