Step-by-Step Guidance for Compound Inequalities (Section 2.5)

Q2. Solve the compound inequality: \( \frac{1}{7}x + 1 \geq 3 \) and \( \frac{1}{5}x - 1 \geq 3 \)

Background

Topic: Compound Inequalities ("and" statements)

This question tests your ability to solve two inequalities simultaneously and find the values of \( x \) that satisfy both conditions at the same time.

Key Terms and Formulas

• Compound Inequality ("and"): Both inequalities must be true at the same time. The solution is the intersection of the two individual solution sets.

• Solving Linear Inequalities: Use inverse operations to isolate \( x \) just as you would in equations, but remember to reverse the inequality sign if you multiply or divide by a negative number.

Step-by-Step Guidance

1. Start by solving each inequality separately.

2. For the first inequality: \( \frac{1}{7}x + 1 \geq 3 \), subtract 1 from both sides to isolate the term with \( x \).

   \( \frac{1}{7}x + 1 - 1 \geq 3 - 1 \)

   \( \frac{1}{7}x \geq 2 \)

3. Multiply both sides by 7 to solve for \( x \) in the first inequality.

   \( x \geq 14 \)

4. For the second inequality: \( \frac{1}{5}x - 1 \geq 3 \), add 1 to both sides to isolate the term with \( x \).

   \( \frac{1}{5}x - 1 + 1 \geq 3 + 1 \)

   \( \frac{1}{5}x \geq 4 \)

5. Multiply both sides by 5 to solve for \( x \) in the second inequality.

   \( x \geq 20 \)

6. Since this is an "and" compound inequality, the solution is the intersection of the two solution sets you found above. Think about which values of \( x \) satisfy both conditions at the same time.

Try solving on your own before revealing the answer!

Q3. Solve the compound inequality and write the solution set in interval notation: \( \frac{1}{3} \leq \frac{4}{3}x + 2 \leq 3 \)

Background

Topic: Compound Inequalities ("sandwich" or "triple" inequalities)

This question tests your ability to solve a compound inequality where \( x \) is "sandwiched" between two bounds. You must isolate \( x \) in the middle by performing the same operation on all three parts.

Key Terms and Formulas

• Compound (Triple) Inequality: An inequality of the form \( a \leq bx + c \leq d \). Solve by performing operations on all three parts simultaneously.

• Interval Notation: Expresses the solution set as an interval, e.g., \( [a, b] \) for all \( x \) between \( a \) and \( b \), inclusive.

Step-by-Step Guidance

1. Start with the compound inequality: \( \frac{1}{3} \leq \frac{4}{3}x + 2 \leq 3 \).

2. Subtract 2 from all three parts to begin isolating \( x \).

   \( \frac{1}{3} - 2 \leq \frac{4}{3}x + 2 - 2 \leq 3 - 2 \)

   \( \frac{1}{3} - 2 \leq \frac{4}{3}x \leq 1 \)

3. Simplify \( \frac{1}{3} - 2 \) to get a single fraction.

   \( \frac{1}{3} - 2 = \frac{1 - 6}{3} = \frac{-5}{3} \)

   So, \( \frac{-5}{3} \leq \frac{4}{3}x \leq 1 \)

4. Divide all three parts by \( \frac{4}{3} \) to solve for \( x \). Remember, dividing by a positive number does not reverse the inequality signs.

   \( \frac{-5}{3} \div \frac{4}{3} \leq x \leq 1 \div \frac{4}{3} \)

5. Simplify each side: dividing by a fraction is the same as multiplying by its reciprocal. Set up the expressions for the endpoints of the interval.

Try solving on your own before revealing the answer!