Compound Inequalities

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Answer Key & Explanations

Compound Inequalities · Grade 9 · Worksheet 2

1. A number line is drawn with points labeled at -10, 0, and 10. Two rays are shown: one starts at -3 and goes to the right with a closed dot, and another starts at 7 and goes to the left with an open dot. These two rays overlap on the number line. Write and solve a compound inequality that represents the overlapping region, then describe the solution set in interval notation. Answer: [-3, 7) Solution: The ray starting at -3 with a closed dot going to the right means x >= -3. The ray starting at 7 with an open dot going to the left means x < 7.
   Full step-by-step solution

   Step 1: The ray starting at -3 with a closed dot going to the right means x >= -3. Step 2: The ray starting at 7 with an open dot going to the left means x < 7. Step 3: The overlapping region is where both conditions are true: x >= -3 AND x < 7. Step 4: This compound inequality can be written as -3 <= x < 7. Step 5: In interval notation, the solution set is [-3, 7), where the bracket [ means -3 is included and the parenthesis ) means 7 is not included. The answer is [-3, 7).

2. Solve: 4x - 9 > 15 OR 2x + 7 < -11 Answer: x > 6 or x < -9 Solution: Solve the first inequality 4x - 9 > 15 Add 9 to both sides: 4x > 24 Divide both sides by 4: x > 6 Solve the second inequality 2x + 7 < -11 Subtract 7 from both sides: 2x < -18 Divide both sides by 2: x < -9 The solution is x > 6 or x < -9.
   Full step-by-step solution

   Step 1: Solve the first inequality 4x - 9 > 15 Add 9 to both sides: 4x > 24 Divide both sides by 4: x > 6 Step 2: Solve the second inequality 2x + 7 < -11 Subtract 7 from both sides: 2x < -18 Divide both sides by 2: x < -9 Step 3: Combine the solutions using OR The solution is x > 6 or x < -9. Step 4: Graph the solution on a number line Draw an open circle at 6 and shade to the right. Draw an open circle at -9 and shade to the left. The two shaded regions are separate, showing the OR condition. The answer is x > 6 or x < -9.

3. Hana is analyzing a compound inequality represented on a number line. The graph shows a ray starting at an open circle at 2 and extending to the left, AND a ray starting at a closed circle at 8 and extending to the right. What compound inequality does this graph represent? Write your answer in the form x < a OR x ≥ b (use appropriate inequality symbols). Answer: x < 2 OR x ≥ 8 Solution: Identify the first ray. It starts at an open circle at 2 and extends to the left. This means all numbers less than 2 are included, but 2 itself is not included.
   Full step-by-step solution

   Step 1: Identify the first ray. It starts at an open circle at 2 and extends to the left. This means all numbers less than 2 are included, but 2 itself is not included. This is written as x < 2. Step 2: Identify the second ray. It starts at a closed circle at 8 and extends to the right. This means all numbers greater than or equal to 8 are included, and 8 itself is included. This is written as x ≥ 8. Step 3: The word "AND" connecting the two rays on the graph means both conditions must be true. However, since no number can be both less than 2 AND greater than or equal to 8 at the same time, the graph actually represents an OR compound inequality. The solution set is the union of the two separate intervals: x < 2 OR x ≥ 8. The answer is x < 2 OR x ≥ 8.

4. Tane is analyzing a compound inequality shown on a number line. The graph shows a closed circle at 9 and an arrow pointing to the right, combined with an open circle at 21 and an arrow pointing to the left. The two regions overlap between the circles. What compound inequality is represented by this graph? Answer: x >= 9 AND x < 21 Solution: Identify the left boundary. The closed circle at 9 means 9 is included, so the inequality is x >= 9. Identify the right boundary.
   Full step-by-step solution

   Step 1: Identify the left boundary. The closed circle at 9 means 9 is included, so the inequality is x >= 9. Step 2: Identify the right boundary. The open circle at 21 means 21 is not included, so the inequality is x < 21. Step 3: Since the graph shows an AND compound inequality (the overlap of both arrows), combine the two inequalities: x >= 9 AND x < 21. Step 4: This can also be written in compact form as 9 <= x < 21. The answer is x >= 9 AND x < 21.

5. Liam is designing a rectangular garden for his school's environmental club. The length of the garden must be at least 5 feet more than twice its width, but no more than 3 feet less than three times its width. If the width is 8 feet, write a compound inequality that represents the possible values for the length of the garden. Answer: 21 ≤ L ≤ 21 Solution: - Length \( L \) must be at least 5 feet more than twice the width \( W \) → \( L \ge 2W + 5 \) - Length \( L \) must be no more than 3 feet less than three times the width \( W \) "3 feet less than three times its width" means \( 3W - 3 \) So: \( L \le 3W - 3 \) 2W + 5 \le L \le 3W - 3…
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We are told: - Length \( L \) must be **at least 5 feet more than twice the width \( W \)** → \( L \ge 2W + 5 \) - Length \( L \) must be **no more than 3 feet less than three times the width \( W \)** "No more than" means \( \le \) "3 feet less than three times its width" means \( 3W - 3 \) So: \( L \le 3W - 3 \) --- **Step 2: Write the compound inequality** From the two conditions: \[ 2W + 5 \le L \le 3W - 3 \] --- **Step 3: Substitute \( W = 8 \)** Lower bound: \( 2(8) + 5 = 16 + 5 = 21 \) Upper bound: \( 3(8) - 3 = 24 - 3 = 21 \) So: \[ 21 \le L \le 21 \] --- **Step 4: Interpret the result** This means \( L \) must be exactly 21 feet. The compound inequality is: \[ 21 \le L \le 21 \] --- **Final answer:** 21 ≤ L ≤ 21

6. A chemical reaction requires the temperature T (in degrees Celsius) to be maintained within a specific range. The reaction proceeds optimally when the temperature is at least 15°C above room temperature but no more than 40°C above room temperature. If room temperature is 22°C, write a compound inequality that represents all acceptable temperature values for this chemical reaction. Answer: 37 ≤ T ≤ 62 Solution: Room temperature is 22°C 'At least 15°C above room temperature' means T ≥ 22 + 15 = 37 'No more than 40°C above room temperature' means T ≤ 22 + 40 = 62 Combine both conditions: 37 ≤ T ≤ 62 The answer is 37 ≤ T ≤ 62.
   Full step-by-step solution

   Step 1: Room temperature is 22°C Step 2: 'At least 15°C above room temperature' means T ≥ 22 + 15 = 37 Step 3: 'No more than 40°C above room temperature' means T ≤ 22 + 40 = 62 Step 4: Combine both conditions: 37 ≤ T ≤ 62 The answer is 37 ≤ T ≤ 62.

7. |3x - 6| + 4 ≤ 13 = ? Answer: -1 ≤ x ≤ 5 Solution: Subtract 4 from both sides: |3x - 6| ≤ 9 Split into two inequalities: 3x - 6 ≤ 9 and 3x - 6 ≥ -9 Solve first inequality: 3x ≤ 15 → x ≤ 5 Solve second inequality: 3x ≥ -3 → x ≥ -1 Combine solutions: -1 ≤ x ≤ 5 The answer is -1 ≤ x ≤ 5.
   Full step-by-step solution

   Step 1: Subtract 4 from both sides: |3x - 6| ≤ 9 Step 2: Split into two inequalities: 3x - 6 ≤ 9 and 3x - 6 ≥ -9 Step 3: Solve first inequality: 3x ≤ 15 → x ≤ 5 Step 4: Solve second inequality: 3x ≥ -3 → x ≥ -1 Step 5: Combine solutions: -1 ≤ x ≤ 5 The answer is -1 ≤ x ≤ 5.