Compound Inequalities

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Compound Inequalities · Grade 9 · Worksheet 3

1. Kaia is a materials scientist developing a new alloy. The alloy's tensile strength S (in MPa) must satisfy two safety conditions for it to be used in aerospace applications: the tensile strength must be at least 16 MPa more than four times the carbon content C (in percent), but cannot exceed 30 MPa more than three times the carbon content. If the carbon content is fixed at 8%, write and solve a compound inequality to determine the acceptable range of tensile strengths for this alloy. Answer: 48 ≤ S ≤ 54 Solution: Write the first condition. The tensile strength must be at least 16 MPa more than four times the carbon content. With carbon content C = 8%, four times the carbon content is 4 × 8 = 32.
   Full step-by-step solution

   Step 1: Write the first condition. The tensile strength must be at least 16 MPa more than four times the carbon content. With carbon content C = 8%, four times the carbon content is 4 × 8 = 32. So S must be at least 32 + 16 = 48. This gives S ≥ 48. Step 2: Write the second condition. The tensile strength cannot exceed 30 MPa more than three times the carbon content. Three times the carbon content is 3 × 8 = 24. So S cannot exceed 24 + 30 = 54. This gives S ≤ 54. Step 3: Combine both conditions with 'and'. Since S must satisfy both conditions, we write: S ≥ 48 AND S ≤ 54. Step 4: Write the compound inequality. This can be written as 48 ≤ S ≤ 54. Step 5: Interpret the solution. The tensile strength must be between 48 MPa and 54 MPa inclusive. The answer is 48 ≤ S ≤ 54.

2. A pharmaceutical company is testing a new medication where the effective dosage range follows a compound inequality. The minimum effective dose is 2.5 mg per kg of body weight, and the maximum safe dose is 7.5 mg per kg. If a patient weighs w kilograms, write the compound inequality that represents all safe and effective dosages d (in mg) for this patient. Answer: 2.5w ≤ d ≤ 7.5w Solution: In medication dosing, compound inequalities often represent therapeutic windows where the lower bound ensures the drug is effective and the upper bound prevents toxicity.
   Full step-by-step solution

   In medication dosing, compound inequalities often represent therapeutic windows where the lower bound ensures the drug is effective and the upper bound prevents toxicity. The dosage is typically proportional to body weight, so both bounds scale with the patient's weight variable.

3. 2x² - 8x - 10 = 0 Answer: x = 5, x = -1 Solution: 2x² - 8x - 10 = 0 Notice that all coefficients (2, -8, -10) are divisible by 2.
   Full step-by-step solution

   Let's solve the equation step by step. We have: 2x² - 8x - 10 = 0 --- **Step 1: Simplify the equation if possible** Notice that all coefficients (2, -8, -10) are divisible by 2. Divide the entire equation by 2: (2x²)/2 - (8x)/2 - 10/2 = 0/2 x² - 4x - 5 = 0 --- **Step 2: Factor the quadratic** We need two numbers that multiply to -5 (the constant term) and add to -4 (the coefficient of x). Possible pairs for -5: 1 and -5 → 1 + (-5) = -4 ✔ This works. So: x² - 4x - 5 = (x - 5)(x + 1) = 0 --- **Step 3: Apply the zero-product property** If (x - 5)(x + 1) = 0, then: x - 5 = 0 or x + 1 = 0 --- **Step 4: Solve each equation** x - 5 = 0 → x = 5 x + 1 = 0 → x = -1 --- **Final Answer:** x = 5, x = -1

4. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). A circle is inscribed inside the triangle, tangent to all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: AB = 6 (horizontal leg) AC = 8 (vertical leg) BC = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 (hypotenuse) Area = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units Perimeter = 6 + 8 + 10 = 24 units Inradius (r) = (2 × Area) / Perimeter r = (2 × 24) / 24 = 48 / 24 = 2 The radius of…
   Full step-by-step solution

   Step 1: Identify the side lengths of the right triangle AB = 6 (horizontal leg) AC = 8 (vertical leg) BC = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 (hypotenuse) Step 2: Calculate the area of the triangle Area = (1/2) × base × height = (1/2) × 6 × 8 = 24 square units Step 3: Calculate the perimeter of the triangle Perimeter = 6 + 8 + 10 = 24 units Step 4: Use the formula for the inradius of a triangle Inradius (r) = (2 × Area) / Perimeter r = (2 × 24) / 24 = 48 / 24 = 2 Step 5: The radius of the inscribed circle is 2 units.

5. Kaia is analyzing a compound inequality on a number line. The shaded region shows all numbers that satisfy both conditions: the point is to the right of -3 (not including -3) and to the left of 7 (not including 7). Write the compound inequality that describes this shaded region, solve for x in terms of a compound inequality involving x, and describe how the graph would look. Answer: -3 < x < 7 Solution: The shaded region is to the right of -3, meaning x is greater than -3. Since -3 is not included, we use a strict inequality: x > -3. The shaded region is to the left of 7, meaning x is less than 7.
   Full step-by-step solution

   Step 1: The shaded region is to the right of -3, meaning x is greater than -3. Since -3 is not included, we use a strict inequality: x > -3. Step 2: The shaded region is to the left of 7, meaning x is less than 7. Since 7 is not included, we use a strict inequality: x < 7. Step 3: Because the region satisfies both conditions simultaneously (AND), we combine them into a compound inequality: -3 < x < 7. Step 4: To describe the graph: Draw a number line. Place an open circle at -3 and an open circle at 7. Shade the entire line segment between -3 and 7. This shows all numbers greater than -3 and less than 7. The answer is -3 < x < 7.

6. A pharmaceutical company is testing a new medication and needs to maintain the drug concentration in a patient's bloodstream between 15 mg/L and 45 mg/L. The concentration C(t) after t hours is modeled by the quadratic function C(t) = -2t² + 20t + 10. Determine the time interval during which the drug concentration remains within the acceptable range. Answer: between 1 and 7 hours Solution: Set up the inequalities: 15 ≤ -2t² + 20t + 10 ≤ 45 Solve the left inequality: 15 ≤ -2t² + 20t + 10 15 - 10 ≤ -2t² + 20t 5 ≤ -2t² + 20t 2t² - 20t + 5 ≤ 0 Solve the right inequality: -2t² + 20t + 10 ≤ 45 -2t² + 20t + 10 - 45 ≤ 0 -2t² + 20t - 35 ≤ 0 Multiply by -1: 2t² - 20t + 35 ≥ 0 For left…
   Full step-by-step solution

   Step 1: Set up the inequalities: 15 ≤ -2t² + 20t + 10 ≤ 45 Step 2: Solve the left inequality: 15 ≤ -2t² + 20t + 10 15 - 10 ≤ -2t² + 20t 5 ≤ -2t² + 20t 2t² - 20t + 5 ≤ 0 Step 3: Solve the right inequality: -2t² + 20t + 10 ≤ 45 -2t² + 20t + 10 - 45 ≤ 0 -2t² + 20t - 35 ≤ 0 Multiply by -1: 2t² - 20t + 35 ≥ 0 Step 4: Find critical points by solving equations: For left boundary: -2t² + 20t + 10 = 15 → -2t² + 20t - 5 = 0 → t = [ -20 ± sqrt(400 - 40) ] / (-4) = [ -20 ± sqrt(360) ] / (-4) = [ -20 ± 6√10 ] / (-4) ≈ 0.26 and 9.74 For right boundary: -2t² + 20t + 10 = 45 → -2t² + 20t - 35 = 0 → t = [ -20 ± sqrt(400 - 280) ] / (-4) = [ -20 ± sqrt(120) ] / (-4) = [ -20 ± 2√30 ] / (-4) ≈ 1 and 7 Step 5: Test intervals: The parabola opens downward (coefficient of t² is negative), so the concentration is between 15 and 45 mg/L when t is between approximately 1 and 7 hours. Step 6: Verify: At t = 1, C(1) = -2(1)² + 20(1) + 10 = 28 mg/L At t = 7, C(7) = -2(49) + 140 + 10 = 12 mg/L (slightly below 15, so exactly at the boundary) The acceptable time interval is between 1 and 7 hours.

7. Solve: 4x - 11 > 13 OR 3x + 7 ≤ -14. Answer: x > 6 or x ≤ -7 Solution: Solve the first inequality: 4x - 11 > 13 Add 11 to both sides: 4x > 24 Divide both sides by 4: x > 6 Solve the second inequality: 3x + 7 ≤ -14 Subtract 7 from both sides: 3x ≤ -21 Divide both sides by 3: x ≤ -7 Combine the solutions with OR: x > 6 or x ≤ -7 - Draw an open circle at 6 and shade…
   Full step-by-step solution

   Step 1: Solve the first inequality: 4x - 11 > 13 Add 11 to both sides: 4x > 24 Divide both sides by 4: x > 6 Step 2: Solve the second inequality: 3x + 7 ≤ -14 Subtract 7 from both sides: 3x ≤ -21 Divide both sides by 3: x ≤ -7 Step 3: Combine the solutions with OR: x > 6 or x ≤ -7 Step 4: Graph the solution on a number line: - Draw an open circle at 6 and shade to the right (for x > 6) - Draw a closed circle at -7 and shade to the left (for x ≤ -7) - The two shaded regions are separate, showing the OR condition. The answer is x > 6 or x ≤ -7.