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Difficulty: HardLinear and Quadratic Inequalities

How many integer values of xx satisfy both the linear inequality 3−2x5≥x+42\frac{3 - 2x}{5} \ge \frac{x + 4}{2} and the quadratic inequality x2+4x−5≤0x^2 + 4x - 5 \le 0?

  1. 4Answer
  2. B
    3
  3. C
    5
  4. D
    2

Answer

4
Solving the linear inequality yields x≤−149≈−1.56x \le -\frac{14}{9} \approx -1.56. Solving the quadratic inequality gives −5≤x≤1-5 \le x \le 1. The overlap between both sets is −5≤x≤−149-5 \le x \le -\frac{14}{9}. The integers falling within this interval are −5,−4,−3-5, -4, -3, and −2-2, making 4 valid integer solutions in total.

Step-by-Step Solution

1
Solve the linear inequality 3−2x5≥x+42\frac{3 - 2x}{5} \ge \frac{x + 4}{2}.
2(3−2x)≥5(x+4)  ⟹  6−4x≥5x+20  ⟹  −9x≥14  ⟹  x≤−149≈−1.562(3 - 2x) \ge 5(x + 4) \implies 6 - 4x \ge 5x + 20 \implies -9x \ge 14 \implies x \le -\frac{14}{9} \approx -1.56.
Clear denominators by multiplying by 10 and reverse the inequality sign when dividing both sides by −9-9.
2
Solve the quadratic inequality x2+4x−5≤0x^2 + 4x - 5 \le 0.
(x+5)(x−1)≤0  ⟹  −5≤x≤1(x + 5)(x - 1) \le 0 \implies -5 \le x \le 1.
Factorize the quadratic expression to find critical points at x=−5x = -5 and x=1x = 1. The region where the product is non-positive is between the roots.
3
Determine the intersection of the two solution sets.
−5≤x≤−149-5 \le x \le -\frac{14}{9}.
Combine the conditions x≤−1.56x \le -1.56 and −5≤x≤1-5 \le x \le 1 to find the set of values satisfying both inequalities simultaneously.
4
Count the integer values within the intersection set [−5,−1.56][-5, -1.56].
The integers are −5,−4,−3,−2-5, -4, -3, -2, giving a total of 4 integers.
Identify all whole numbers within the combined solution interval.

Key Concept

Simultaneous Linear and Quadratic Inequalities
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Topics

Linear and Quadratic Inequalities

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