Create Inequalities

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Answer Key & Explanations

Create Inequalities · Grade 9 · Worksheet 1

1. 2x² - 8x + 6 ≤ 0 Answer: 1 ≤ x ≤ 3 Solution: 2x² - 8x + 6 ≤ 0 All terms are divisible by 2: 2(x² - 4x + 3) ≤ 0 Divide both sides by 2 (since 2 > 0, the inequality sign stays the same): x² - 4x + 3 ≤ 0 We look for two numbers that multiply to 3 and add to -4.
   Full step-by-step solution

   Let's solve the inequality step-by-step. We have: 2x² - 8x + 6 ≤ 0 --- **Step 1: Factor out the common factor** All terms are divisible by 2: 2(x² - 4x + 3) ≤ 0 Divide both sides by 2 (since 2 > 0, the inequality sign stays the same): x² - 4x + 3 ≤ 0 --- **Step 2: Factor the quadratic** We look for two numbers that multiply to 3 and add to -4. Those numbers are -1 and -3. So: x² - 4x + 3 = (x - 1)(x - 3) Thus: (x - 1)(x - 3) ≤ 0 --- **Step 3: Find the zeros** Set (x - 1)(x - 3) = 0 x - 1 = 0 → x = 1 x - 3 = 0 → x = 3 These are the points where the expression equals zero. --- **Step 4: Determine intervals** The zeros split the number line into three intervals: (-∞, 1), (1, 3), (3, ∞) We test each interval to see where (x - 1)(x - 3) is negative or positive. - For x < 1, pick x = 0: (0 - 1)(0 - 3) = (-1)(-3) = 3 > 0 → positive - For 1 < x < 3, pick x = 2: (2 - 1)(2 - 3) = (1)(-1) = -1 < 0 → negative - For x > 3, pick x = 4: (4 - 1)(4 - 3) = (3)(1) = 3 > 0 → positive --- **Step 5: Apply the inequality ≤ 0** We want (x - 1)(x - 3) ≤ 0. That means we want where it is negative or zero. It is zero at x = 1 and x = 3. It is negative between 1 and 3. So the solution is: 1 ≤ x ≤ 3 --- **Final answer:** 1 ≤ x ≤ 3

2. Aisha is designing a rectangular vegetable patch with a fixed perimeter of 48 meters. She wants the length to be at least 3 meters more than three times the width. If w represents the width in meters, write and solve an inequality to determine all possible widths that satisfy Aisha's design requirements. Answer: w ≤ 5.25 Solution: The perimeter P of a rectangle is given by P = 2l + 2w, where l is the length and w is the width. We are told the perimeter is 48 meters. So, 2l + 2w = 48.
   Full step-by-step solution

   Step 1: The perimeter P of a rectangle is given by P = 2l + 2w, where l is the length and w is the width. We are told the perimeter is 48 meters. Step 2: So, 2l + 2w = 48. Divide the entire equation by 2 to simplify: l + w = 24. Therefore, the length can be expressed as l = 24 - w. Step 3: The problem states that the length must be at least 3 meters more than three times the width. 'At least' means greater than or equal to. This gives the inequality: l ≥ 3w + 3. Step 4: Substitute the expression for l from Step 2 (l = 24 - w) into the inequality from Step 3: 24 - w ≥ 3w + 3. Step 5: Solve the inequality for w. First, add w to both sides: 24 ≥ 4w + 3. Step 6: Subtract 3 from both sides: 21 ≥ 4w. Step 7: Divide both sides by 4: 21/4 ≥ w, which simplifies to w ≤ 5.25. Step 8: Since width must be a positive measurement, the final solution is 0 < w ≤ 5.25.

3. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length to be at least 5 meters more than the width. Write an inequality that represents all possible widths (w) in meters for Liam's garden that satisfy this condition. Answer: w ≤ 7.5 Solution: Write down what we know. P = 2 × length + 2 × width Given: P = 40 meters Let width = w meters Let length = L meters 2L + 2w = 40 Simplify the perimeter equation.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Write down what we know.** Perimeter of a rectangle: P = 2 × length + 2 × width Given: P = 40 meters Let width = w meters Let length = L meters So: 2L + 2w = 40 --- **Step 2: Simplify the perimeter equation.** Divide both sides by 2: L + w = 20 So: L = 20 − w --- **Step 3: Apply the length condition.** The problem says: length is at least 5 meters more than the width. "At least" means ≥. So: L ≥ w + 5 --- **Step 4: Substitute L from Step 2 into the inequality.** 20 − w ≥ w + 5 --- **Step 5: Solve for w.** 20 − w ≥ w + 5 20 − 5 ≥ w + w 15 ≥ 2w So: 2w ≤ 15 w ≤ 15/2 w ≤ 7.5 --- **Step 6: Interpret the result.** The width must be positive (w > 0), but the main constraint from the condition is w ≤ 7.5. So possible widths: 0 < w ≤ 7.5 meters. --- **Final answer:** w ≤ 7.5

4. 2x² - 12x + 16 > 0 Answer: x < 2 or x > 4 Solution: Step 1: Set the quadratic equal to zero to find the roots: 2x² - 12x + 16 = 0 Step 2: Divide by 2 to simplify: x² - 6x + 8 = 0 Step 3: Factor the quadratic: (x - 2)(x - 4) = 0 Step 4: Solve for x: x = 2 or x = 4 Step 5: These roots divide the number line into three intervals: (-∞, 2), (2, 4),…
   Full step-by-step solution

   Step 1: Set the quadratic equal to zero to find the roots: 2x² - 12x + 16 = 0 Step 2: Divide by 2 to simplify: x² - 6x + 8 = 0 Step 3: Factor the quadratic: (x - 2)(x - 4) = 0 Step 4: Solve for x: x = 2 or x = 4 Step 5: These roots divide the number line into three intervals: (-∞, 2), (2, 4), and (4, ∞) Step 6: Test a point in (-∞, 2): x = 0 gives 2(0)² - 12(0) + 16 = 16 > 0 ✓ Step 7: Test a point in (2, 4): x = 3 gives 2(9) - 36 + 16 = 18 - 36 + 16 = -2 < 0 ✗ Step 8: Test a point in (4, ∞): x = 5 gives 2(25) - 60 + 16 = 50 - 60 + 16 = 6 > 0 ✓ Step 9: Since we have > (not ≥), the solution is x < 2 or x > 4

5. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length of the garden to be at least 5 meters more than twice its width. Write and solve an inequality to determine the possible widths (in meters) that satisfy Liam's design requirements. Answer: w ≤ 5 Solution: w = width of the garden (in meters) L = length of the garden (in meters) Perimeter of a rectangle = 2(L + w) Given perimeter = 40 m 2(L + w) = 40 Divide both sides by 2: L + w = 20 L = 20 - w The problem says: length is at least 5 meters more than twice the width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let w = width of the garden (in meters) L = length of the garden (in meters) --- **Step 2: Use the perimeter information** Perimeter of a rectangle = 2(L + w) Given perimeter = 40 m So: 2(L + w) = 40 Divide both sides by 2: L + w = 20 Thus: L = 20 - w --- **Step 3: Translate the condition on length** The problem says: length is at least 5 meters more than twice the width. "At least" means ≥. Twice the width = 2w 5 more than twice the width = 2w + 5 So: L ≥ 2w + 5 --- **Step 4: Substitute L from Step 2 into the inequality** L = 20 - w So: 20 - w ≥ 2w + 5 --- **Step 5: Solve the inequality** 20 - w ≥ 2w + 5 Subtract 20 from both sides: -w ≥ 2w - 15 Subtract 2w from both sides: -w - 2w ≥ -15 -3w ≥ -15 Divide both sides by -3 (remember: dividing by a negative flips the inequality sign): w ≤ 5 --- **Step 6: Interpret the result** The width must be positive (w > 0), so: 0 < w ≤ 5 But the problem only asks for the inequality from the given condition, which is w ≤ 5. --- **Final answer:** w ≤ 5

6. Aisha is designing a rectangular banner for a school event. The banner's length must be at least 3 meters more than twice its width. Due to space constraints, the area of the banner cannot exceed 44 square meters. If w represents the width of the banner in meters, which inequality represents all possible width values that satisfy both conditions? Answer: B. 2w^2 + 3w - 44 ≤ 0 Solution: Express the length in terms of width. The length is at least 3 more than twice the width: length = 2w + 3 Write the area formula: Area = length × width = (2w + 3) × w = 2w^2 + 3w Apply the area constraint: Area ≤ 44, so 2w^2 + 3w ≤ 44 Rearrange to standard quadratic inequality form: 2w^2 + 3w -…
   Full step-by-step solution

   Step 1: Express the length in terms of width. The length is at least 3 more than twice the width: length = 2w + 3 Step 2: Write the area formula: Area = length × width = (2w + 3) × w = 2w^2 + 3w Step 3: Apply the area constraint: Area ≤ 44, so 2w^2 + 3w ≤ 44 Step 4: Rearrange to standard quadratic inequality form: 2w^2 + 3w - 44 ≤ 0 Step 5: The correct inequality is 2w^2 + 3w - 44 ≤ 0, which corresponds to choice C.

7. Mere is organizing a school fundraiser. She needs to raise at least $600. She already has $120 from donations and plans to sell tickets for $8 each. Write an inequality to represent the number of tickets, t, she must sell to meet her goal. Then solve for t. Answer: t ≥ 60 Solution: Let t represent the number of tickets sold. Money from tickets = 8t. Total money raised = 120 + 8t.
   Full step-by-step solution

   Step 1: Let t represent the number of tickets sold. Step 2: Money from tickets = 8t. Step 3: Total money raised = 120 + 8t. Step 4: She needs at least $600, so the inequality is 120 + 8t ≥ 600. Step 5: Subtract 120 from both sides: 8t ≥ 480. Step 6: Divide both sides by 8: t ≥ 60. Step 7: Since t must be a whole number, the minimum number of tickets is 60. Final answer: t ≥ 60.