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Which point would be a solution to the system of linear inequalities shown below?

y, is less than or equal to, minus, start fraction, 2, divided by, 3, end fraction, x, minus, 2, y, is greater than or equal to, minus, one half, x, minus, 1
y≤−
3
2
​
x−2y≥−
2
1
​
x−1
Answer
Multiple Choice Answers
left parenthesis, 6, comma, minus, 5, right parenthesis(6,−5)
left parenthesis, 0, comma, minus, 1, right parenthesis(0,−1)
left parenthesis, minus, 6, comma, minus, 7, right parenthesis(−6,−7)
left parenthesis, minus, 12, comma, 5, right parenthesis(−12,5)

1 answer

To determine which point is a solution to the given system of linear inequalities, we will evaluate each of the given points against both inequalities:

  1. \( y \leq -\frac{2}{3}x - 2 \)

  2. \( -2y \geq -\frac{1}{2}x - 1 \) (which can be rewritten as \( y \leq \frac{1}{4}x + \frac{1}{2} \))

Now let's evaluate each point:

Option A: \( (6, -5) \)

  1. \( -5 \leq -\frac{2}{3}(6) - 2 \)
    • Calculating right side: \[ -\frac{2}{3}(6) = -4 \] \[ -4 - 2 = -6 \]
    • So, the inequality becomes \( -5 \leq -6 \) (FALSE)

Option B: \( (0, -1) \)

  1. \( -1 \leq -\frac{2}{3}(0) - 2 \)
    • Right side: \[ -0 - 2 = -2 \]
    • So, the inequality becomes \( -1 \leq -2 \) (FALSE)

Option C: \( (-6, -7) \)

  1. \( -7 \leq -\frac{2}{3}(-6) - 2 \)

    • Right side: \[ -\frac{2}{3}(-6) = 4 \] \[ 4 - 2 = 2 \]
    • So, the inequality becomes \( -7 \leq 2 \) (TRUE)

  2. \(-2(-7) \geq -\frac{1}{2}(-6) - 1\)

    • Left side: \( 14 \)
    • Right side: \[ -\frac{1}{2}(-6) - 1 = 3 - 1 = 2 \]
    • So, the inequality becomes \( 14 \geq 2 \) (TRUE)

Option D: \( (-12, 5) \)

  1. \( 5 \leq -\frac{2}{3}(-12) - 2 \)

    • Right side: \[ -\frac{2}{3}(-12) = 8 \] \[ 8 - 2 = 6 \]
    • So, the inequality becomes \( 5 \leq 6 \) (TRUE)

  2. \(-2(5) \geq -\frac{1}{2}(-12) - 1\)

    • Left side: \( -10 \)
    • Right side: \[ -\frac{1}{2}(-12) - 1 = 6 - 1 = 5 \]
    • So, the inequality becomes \( -10 \geq 5 \) (FALSE)

Conclusion

The only point that satisfies both inequalities is \( (-6, -7) \).

The correct answer is: \( \text{Option C: } (-6, -7) \).