Compound Inequalities

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Compound Inequalities · Grade 9 · Worksheet 1

1. Olivia is a marine biologist monitoring the oxygen concentration (in mg/L) in a coastal estuary. The concentration C must satisfy two conditions to support aquatic life: it must be at least 20 mg/L, and it must be less than 50 mg/L. If the concentration is modeled by the expression 5x - 10, where x is the depth in meters, write and solve the compound inequality that represents all depths x for which the oxygen concentration meets both conditions. Answer: 6 ≤ x < 12 Solution: Write the compound inequality. The concentration must be at least 20 mg/L: 5x - 10 ≥ 20. The concentration must be less than 50 mg/L: 5x - 10 < 50.
   Full step-by-step solution

   Step 1: Write the compound inequality. The concentration must be at least 20 mg/L: 5x - 10 ≥ 20. The concentration must be less than 50 mg/L: 5x - 10 < 50. Together: 5x - 10 ≥ 20 AND 5x - 10 < 50. Step 2: Solve the first inequality: 5x - 10 ≥ 20. Add 10 to both sides: 5x ≥ 30. Divide by 5: x ≥ 6. Step 3: Solve the second inequality: 5x - 10 < 50. Add 10 to both sides: 5x < 60. Divide by 5: x < 12. Step 4: Combine the results: x ≥ 6 AND x < 12 means 6 ≤ x < 12. The solution set is all depths from 6 meters up to but not including 12 meters.

2. Solve the compound inequality: 4x - 7 ≤ 9 and 2x + 5 > 11. What is the smallest integer value of x that satisfies both inequalities? Answer: 4 Solution: Step 1: Solve the first inequality: 4x - 7 ≤ 9 Add 7 to both sides: 4x ≤ 16 Divide both sides by 4: x ≤ 4 Step 2: Solve the second inequality: 2x + 5 > 11 Subtract 5 from both sides: 2x > 6 Divide both sides by 2: x > 3 Step 3: Combine the solutions: x ≤ 4 and x > 3 This means x must be greater…
   Full step-by-step solution

   Step 1: Solve the first inequality: 4x - 7 ≤ 9 Add 7 to both sides: 4x ≤ 16 Divide both sides by 4: x ≤ 4 Step 2: Solve the second inequality: 2x + 5 > 11 Subtract 5 from both sides: 2x > 6 Divide both sides by 2: x > 3 Step 3: Combine the solutions: x ≤ 4 and x > 3 This means x must be greater than 3 and less than or equal to 4 Step 4: Find the smallest integer that satisfies both conditions The integers that satisfy both are: 4 Since x > 3, x cannot be 3 Since x ≤ 4, x can be 4 Step 5: Verify x = 4 First inequality: 4(4) - 7 = 16 - 7 = 9 ≤ 9 ✓ Second inequality: 2(4) + 5 = 8 + 5 = 13 > 11 ✓ The smallest integer value is 4.

3. |2x - 5| + 3 ≥ 10 = ? Answer: x ≤ -1 or x ≥ 6 Solution: |2x - 5| + 3 ≥ 10 |2x - 5| ≥ 7 Case 1: 2x - 5 ≥ 7 Case 2: 2x - 5 ≤ -7 Solve Case 1 2x - 5 ≥ 7 2x ≥ 12 x ≥ 6 Solve Case 2 2x - 5 ≤ -7 2x ≤ -2 x ≤ -1 The solution is x ≤ -1 or x ≥ 6
   Full step-by-step solution

   Step 1: Isolate the absolute value expression |2x - 5| + 3 ≥ 10 |2x - 5| ≥ 7 Step 2: Set up two inequalities for the absolute value Case 1: 2x - 5 ≥ 7 Case 2: 2x - 5 ≤ -7 Step 3: Solve Case 1 2x - 5 ≥ 7 2x ≥ 12 x ≥ 6 Step 4: Solve Case 2 2x - 5 ≤ -7 2x ≤ -2 x ≤ -1 Step 5: Combine the solutions The solution is x ≤ -1 or x ≥ 6

4. Solve: 7x - 2 > 12 OR 2x + 7 < -7 Answer: x > 2 or x < -7 Solution: Solve the first inequality: 7x - 2 > 12 Add 2 to both sides: 7x > 14 Divide both sides by 7: x > 2 Solve the second inequality: 2x + 7 < -7 Subtract 7 from both sides: 2x < -14 Divide both sides by 2: x < -7 Combine using OR: The solution is x > 2 or x < -7.
   Full step-by-step solution

   Step 1: Solve the first inequality: 7x - 2 > 12 Add 2 to both sides: 7x > 14 Divide both sides by 7: x > 2 Step 2: Solve the second inequality: 2x + 7 < -7 Subtract 7 from both sides: 2x < -14 Divide both sides by 2: x < -7 Step 3: Combine using OR: The solution is x > 2 or x < -7. In interval notation: (-∞, -7) ∪ (2, ∞) Graph: On a number line, draw an open circle at -7 and shade to the left, and an open circle at 2 and shade to the right.

5. |2x + 3| - 5 ≤ 8 = ? Answer: -8 ≤ x ≤ 5 Solution: Add 5 to both sides: |2x + 3| - 5 + 5 ≤ 8 + 5 → |2x + 3| ≤ 13 For |A| ≤ k, we have -k ≤ A ≤ k.
   Full step-by-step solution

   Step 1: Add 5 to both sides: |2x + 3| - 5 + 5 ≤ 8 + 5 → |2x + 3| ≤ 13 Step 2: For |A| ≤ k, we have -k ≤ A ≤ k. So -13 ≤ 2x + 3 ≤ 13 Step 3: Solve the compound inequality by subtracting 3 from all parts: -13 - 3 ≤ 2x + 3 - 3 ≤ 13 - 3 → -16 ≤ 2x ≤ 10 Step 4: Divide all parts by 2: -16/2 ≤ 2x/2 ≤ 10/2 → -8 ≤ x ≤ 5 The solution is -8 ≤ x ≤ 5.

6. A rectangular garden has a length that is 3 meters more than twice its width. The area of the garden must be between 35 and 60 square meters. Write a compound inequality that represents all possible widths of the garden, then solve it to find the range of possible widths. Answer: 5 < w < 6 Solution: Compound inequalities often arise in optimization problems where a quantity must fall within a specific range.
   Full step-by-step solution

   Compound inequalities often arise in optimization problems where a quantity must fall within a specific range. For geometric applications, we typically express all dimensions in terms of one variable, then use the area formula to create an inequality. The solution involves finding where a quadratic expression lies between two values, which requires solving two separate inequalities and finding their intersection.

7. Kaia is a marine biologist studying the optimal salinity levels for a species of sea turtle. The salinity S (in parts per thousand) in a research tank must satisfy two conditions for the turtles to thrive: the salinity must be at least 21 parts per thousand, and it must be less than 37 parts per thousand. However, due to equipment limitations, the actual salinity S can deviate from the target range by at most 5 parts per thousand in either direction. Write a compound inequality that represents all possible salinity values S that are acceptable, then solve it to find the range of acceptable salinities. Answer: 16 ≤ S < 42 Solution: The target salinity range is at least 21 ppt (S ≥ 21) and less than 37 ppt (S < 37). The equipment allows a deviation of at most 5 ppt in either direction.
   Full step-by-step solution

   Step 1: The target salinity range is at least 21 ppt (S ≥ 21) and less than 37 ppt (S < 37). So the target inequality is 21 ≤ S < 37. Step 2: The equipment allows a deviation of at most 5 ppt in either direction. This means we subtract 5 from the lower bound and add 5 to the upper bound to find the full acceptable range. Step 3: Lower bound: 21 - 5 = 16. Upper bound: 37 + 5 = 42. Step 4: The compound inequality for the acceptable salinity S is: S ≥ 16 AND S < 42, which can be written as 16 ≤ S < 42. Step 5: The solution set in interval notation is [16, 42). The answer is 16 ≤ S < 42.