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which point is a solution to the simultaneous inequalities y >1/2x + 5 and y < -2x + 1? (1, 5.5) (4, 8) (-5, 5) or (-1.6, 4.2)

3 answers

To find which point is a solution to the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will test each point by substituting the coordinates into both inequalities.

  1. For the point (1, 5.5):

    • First inequality: \( y > \frac{1}{2}(1) + 5 \)
      \( 5.5 > 0.5 + 5 \)
      \( 5.5 > 5.5 \) (False)
    • Second inequality: \( y < -2(1) + 1 \)
      \( 5.5 < -2 + 1 \)
      \( 5.5 < -1 \) (False)

  2. For the point (4, 8):

    • First inequality: \( y > \frac{1}{2}(4) + 5 \)
      \( 8 > 2 + 5 \)
      \( 8 > 7 \) (True)
    • Second inequality: \( y < -2(4) + 1 \)
      \( 8 < -8 + 1 \)
      \( 8 < -7 \) (False)

  3. For the point (-5, 5):

    • First inequality: \( y > \frac{1}{2}(-5) + 5 \)
      \( 5 > -2.5 + 5 \)
      \( 5 > 2.5 \) (True)
    • Second inequality: \( y < -2(-5) + 1 \)
      \( 5 < 10 + 1 \)
      \( 5 < 11 \) (True)

  4. For the point (-1.6, 4.2):

    • First inequality: \( y > \frac{1}{2}(-1.6) + 5 \)
      \( 4.2 > -0.8 + 5 \)
      \( 4.2 > 4.2 \) (False)
    • Second inequality: \( y < -2(-1.6) + 1 \)
      \( 4.2 < 3.2 + 1 \)
      \( 4.2 < 4.2 \) (False)

After evaluating all the points, the only point that satisfies both inequalities is:

(-5, 5).

which point is a solution to the simultaneous inequalities y < 1/2x and y > 1/3x + 3? (2, 2), (10, 5), (3, 6), or (25, 12)?

To determine which point satisfies the simultaneous inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), we will evaluate each point by substituting the coordinates into both inequalities.

1. For the point (2, 2):

  • First inequality: \[ 2 < \frac{1}{2}(2) \implies 2 < 1 \quad \text{(False)} \]
  • Second inequality: \[ 2 > \frac{1}{3}(2) + 3 \implies 2 > \frac{2}{3} + 3 \implies 2 > 3.67 \quad \text{(False)} \]

2. For the point (10, 5):

  • First inequality: \[ 5 < \frac{1}{2}(10) \implies 5 < 5 \quad \text{(False)} \]
  • Second inequality: \[ 5 > \frac{1}{3}(10) + 3 \implies 5 > \frac{10}{3} + 3 \implies 5 > 3.33 + 3 \implies 5 > 6.33 \quad \text{(False)} \]

3. For the point (3, 6):

  • First inequality: \[ 6 < \frac{1}{2}(3) \implies 6 < 1.5 \quad \text{(False)} \]
  • Second inequality: \[ 6 > \frac{1}{3}(3) + 3 \implies 6 > 1 + 3 \implies 6 > 4 \quad \text{(True)} \]

4. For the point (25, 12):

  • First inequality: \[ 12 < \frac{1}{2}(25) \implies 12 < 12.5 \quad \text{(True)} \]
  • Second inequality: \[ 12 > \frac{1}{3}(25) + 3 \implies 12 > \frac{25}{3} + 3 \implies 12 > 8.33 + 3 \implies 12 > 11.33 \quad \text{(True)} \]

Conclusion:

The only point that satisfies both inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \) is:

(25, 12).