Math Problem Statement

A system of inequalities is shown below:

y<−2x+1y<−2x+1

x+2y≥−4x+2y≥−4

Which of the following is in the solution set of this system of inequalities?

A.(−9,1)(−9,1)

B.(−5,4)(−5,4)

C.(2,−3)(2,−3)

D.(6,−8)(6,−8)

Solution

To determine which point is in the solution set of this system of inequalities, we need to test each given point against the two inequalities:

  1. y<2x+1y < -2x + 1
  2. x+2y4x + 2y \geq -4

Test Point A: (9,1)(-9, 1)

  • For y<2x+1y < -2x + 1: 1<2(9)+11 < -2(-9) + 1 1<18+11 < 18 + 1 1<19(True)1 < 19 \quad \text{(True)}
  • For x+2y4x + 2y \geq -4: 9+2(1)4-9 + 2(1) \geq -4 9+24-9 + 2 \geq -4 74(False)-7 \geq -4 \quad \text{(False)} Thus, point A does not satisfy both inequalities.

Test Point B: (5,4)(-5, 4)

  • For y<2x+1y < -2x + 1: 4<2(5)+14 < -2(-5) + 1 4<10+14 < 10 + 1 4<11(True)4 < 11 \quad \text{(True)}
  • For x+2y4x + 2y \geq -4: 5+2(4)4-5 + 2(4) \geq -4 5+84-5 + 8 \geq -4 34(True)3 \geq -4 \quad \text{(True)} Thus, point B satisfies both inequalities.

Test Point C: (2,3)(2, -3)

  • For y<2x+1y < -2x + 1: 3<2(2)+1-3 < -2(2) + 1 3<4+1-3 < -4 + 1 3<3(False, because -3 is not less than -3)-3 < -3 \quad \text{(False, because -3 is not less than -3)} Thus, point C does not satisfy the first inequality.

Test Point D: (6,8)(6, -8)

  • For y<2x+1y < -2x + 1: 8<2(6)+1-8 < -2(6) + 1 8<12+1-8 < -12 + 1 8<11(False)-8 < -11 \quad \text{(False)} Thus, point D does not satisfy the first inequality.

Conclusion:

The only point that satisfies both inequalities is B: (5,4)(-5, 4).


Do you need further explanation or any additional details? Here are five related questions to expand your understanding:

  1. How do you graph systems of inequalities on a coordinate plane?
  2. How would the solution set look visually for this system?
  3. How can you determine if a system of inequalities has no solution?
  4. What happens if both inequalities have the same slope?
  5. Can inequalities with different variables (like xx and yy) be solved similarly?

Tip: When testing a point in an inequality, plug the coordinates into both inequalities and check for consistency with the inequality signs.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Systems of Inequalities

Formulas

y < −2x + 1
x + 2y ≥ −4

Theorems

Testing Points in Inequalities
Intersection of Solution Sets

Suitable Grade Level

Grades 8-10