Q1. Solve the following inequality. Write the solution set using interval notation:

19 + 7x \geq 2x - 5

Background

Topic: Linear Inequalities

This question tests your ability to solve a linear inequality and express the solution set using interval notation.

Key Terms and Formulas:

• Linear Inequality: An inequality involving a linear expression, such as $ax + b \leq c$.

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

Step-by-Step Guidance

1. Start by isolating the variable $x$ on one side. Subtract $2x$ from both sides:

   $19 + 7x - 2x \geq -5$

2. Simplify the left side by combining like terms:

   $19 + 5x \geq -5$

3. Subtract $19$ from both sides to further isolate $x$:

   $5x \geq -5 - 19$

4. Simplify the right side and prepare to solve for $x$ by dividing both sides by $5$.

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Q2. Solve the following inequality. Write the solution set in interval notation:

7(x - 3) \leq 2(4x - 1)

Background

Topic: Linear Inequalities with Distribution

This question tests your ability to solve inequalities that require distributing and combining like terms before isolating the variable.

Key Terms and Formulas:

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

• Interval Notation: Used to express the solution set.

Step-by-Step Guidance

1. Apply the distributive property to both sides:

   $7x - 21 \leq 8x - 2$

2. Get all terms involving $x$ on one side and constants on the other. Subtract $7x$ from both sides:

   $-21 \leq x - 2$

3. Add $2$ to both sides to isolate $x$:

   $-21 + 2 \leq x$

4. Simplify the left side and write the solution in $x \geq$ form, then express in interval notation.

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Q3. Solve the following inequality. Write the solution set in interval notation:

-3(x - 1) < -2[5 + 8(x + 5)]

Background

Topic: Linear Inequalities with Distribution and Combining Like Terms

This question tests your ability to handle distribution, combining like terms, and solving inequalities with negative coefficients.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Distribute $-3$ on the left and $-2$ on the right:

   $-3x + 3 < -2 \times 5 - 2 \times 8(x + 5)$

2. Simplify the right side by distributing $-2$ to both $5$ and $8(x + 5)$, then expand $8(x + 5)$.

3. Combine like terms on both sides to get all $x$ terms on one side and constants on the other.

4. Isolate $x$ and prepare to divide by the coefficient of $x$ (remember to reverse the inequality if dividing by a negative).

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Q4. Solve.

\frac{1}{2} + \frac{3}{7} \geq \frac{x}{14}

Background

Topic: Linear Inequalities with Fractions

This question tests your ability to solve inequalities involving fractions and to express the solution in terms of $x$.

Key Terms and Formulas:

• Common Denominator: To add or compare fractions, rewrite them with a common denominator.

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

Step-by-Step Guidance

1. Find a common denominator for the fractions on the left side.

2. Add the fractions together to get a single fraction on the left.

3. Set up the inequality so that you have a single fraction on each side: $\frac{a}{b} \geq \frac{x}{14}$.

4. Multiply both sides by $14$ to clear the denominators and solve for $x$.

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Q5. Solve the following inequality. Write the solution set using interval notation:

3(x - 6) + 3x - 10 \geq 2(x - 8) + 10x

Background

Topic: Linear Inequalities with Distribution and Combining Like Terms

This question tests your ability to distribute, combine like terms, and solve for $x$ in a linear inequality.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Distribute $3$ on the left and $2$ on the right:

   $3x - 18 + 3x - 10 \geq 2x - 16 + 10x$

2. Combine like terms on both sides to simplify the inequality.

3. Get all $x$ terms on one side and constants on the other.

4. Isolate $x$ and prepare to divide by the coefficient of $x$ to solve.

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Q6. Solve the inequality:

4x + 1 < 4(x - 2)

Background

Topic: Linear Inequalities with Distribution

This question tests your ability to solve a linear inequality that requires distributing and isolating the variable.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Distribute $4$ on the right side:

   $4x + 1 < 4x - 8$

2. Subtract $4x$ from both sides to get all $x$ terms on one side:

   $1 < -8$

3. Analyze the resulting statement to determine if there are any solutions or if the solution set is empty.

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Reference Table for Interval Notation and Graphs

This table summarizes how to write solution sets for inequalities using set notation, graph, and interval notation. Use it as a reference when expressing your answers.