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Use the image to answer the question.

A coordinate plane shows a graph of an upward-sloping, solid, diagonal line, with shading extending from the line to the left. The line begins in quadrant 4 and ends in quadrant 1.

Which of the following points is a solution to the linear inequality 4x−y≤10 , as shown on the graph?

Option #1: (2 1/2,−1/2)

Option #2: (2,−3)

Option #3: (3,2)

(1 point)
The point in Option # is a solution to the linear inequality 4x−y≤10.

1 answer

To determine which of the given points is a solution to the inequality \(4x - y \leq 10\), we need to evaluate the inequality for each option.

Option #1: \( (2.5, -0.5) \)

  1. Substitute \(x = 2.5\) and \(y = -0.5\) into the inequality: \[ 4(2.5) - (-0.5) \leq 10 \] \[ 10 + 0.5 \leq 10 \] \[ 10.5 \leq 10 \quad (\text{not true}) \]

Option #2: \( (2, -3) \)

  1. Substitute \(x = 2\) and \(y = -3\) into the inequality: \[ 4(2) - (-3) \leq 10 \] \[ 8 + 3 \leq 10 \] \[ 11 \leq 10 \quad (\text{not true}) \]

Option #3: \( (3, 2) \)

  1. Substitute \(x = 3\) and \(y = 2\) into the inequality: \[ 4(3) - 2 \leq 10 \] \[ 12 - 2 \leq 10 \] \[ 10 \leq 10 \quad (\text{true}) \]

Since only Option #3 satisfies the inequality \(4x - y \leq 10\), the answer is:

The point in Option #3 (3, 2) is a solution to the linear inequality \(4x - y \leq 10\).