Use the image to answer the question../././.

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: (212,−12)

Option #2: (2,−3)

Option #3: (3,2)

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

Answer 1

To determine which point is a solution to the linear inequality \(4x - y \leq 10\), we can substitute each option into the inequality and see if it satisfies the condition.

1. Option #1: (212, -12)
   Substitute \(x = 212\) and \(y = -12\):
   \[ 4(212) - (-12) \leq 10 \]
   \[ 848 + 12 \leq 10 \]
   \[ 860 \leq 10 \] (False)

2. Option #2: (2, -3)
   Substitute \(x = 2\) and \(y = -3\):
   \[ 4(2) - (-3) \leq 10 \]
   \[ 8 + 3 \leq 10 \]
   \[ 11 \leq 10 \] (False)

3. Option #3: (3, 2)
   Substitute \(x = 3\) and \(y = 2\):
   \[ 4(3) - 2 \leq 10 \]
   \[ 12 - 2 \leq 10 \]
   \[ 10 \leq 10 \] (True)

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

Answer 2

Use the image to answer the question.

A dashed line with a negative slope is plotted on a 4-quadrant coordinate plane. The area below the line is shaded. A labeled coordinate point is plotted in the area above the line.

The graph shows the inequality y<−12x−3
. Is the point (−1,4)
part of the solution set? Enter 1 for yes or 2 for no.

(1 point)

Answer 3

To determine whether the point \((-1, 4)\) is part of the solution set for the inequality \(y < -\frac{1}{2}x - 3\), we need to substitute the coordinates of the point into the inequality.

1. Substitute \(x = -1\) and \(y = 4\) into the inequality:

\[ 4 < -\frac{1}{2}(-1) - 3 \]

2. Simplify the right side:

\[ 4 < \frac{1}{2} - 3 \]

\[ 4 < \frac{1}{2} - \frac{6}{2} \]

\[ 4 < -\frac{5}{2} \]

3. This inequality is false since \(4\) is not less than \(-\frac{5}{2}\).

Since the statement is false, the point \((-1, 4)\) is not part of the solution set.

The answer is: 2 (for no).

Answer 4

Use the image to answer the question.

A straight line and a point are plotted on a coordinate plane. The line slopes upward from left to right, starting in quadrant 3 and rising through quadrant 1. The area to the right of the line is shaded, the majority of which is in quadrants 1 and 4.

Which equation matches the inequality in the given graph?

Option #1: y<32x+3


Option #2: y≤32x+3


Option #3: y>32x+3


Option #4: y≥32x+3


(1 point)
The equation that matches the graph is Option #
.

Answer 5

To determine which equation matches the inequality shown in the graph, we can analyze two key aspects based on its description:

1. Slope Direction: The line slopes upward from left to right, indicating that it has a positive slope. This suggests that any inequality representing the region above the line would have a "greater than" (\(>\) or \(\geq\)) relationship, while the region below the line would have a "less than" (\(<\) or \(\leq\)) relationship.

2. Shaded Region: The area to the right of the line is shaded, which typically indicates that the solution set includes values greater than the line.

Given this reasoning, we can conclude:

• The line itself could be represented by the equation \(y = \frac{3}{2}x + 3\).
• Since the area to the right of the line is shaded, and the line itself is included (as it appears no dotted or dashed line is mentioned), we will choose the option with the \( \geq \) symbol.

Based on the provided options:

• Option #1: \(y < \frac{3}{2}x + 3\) (less than) - Does not match.
• Option #2: \(y \leq \frac{3}{2}x + 3\) (less than or equal to) - Does not match.
• Option #3: \(y > \frac{3}{2}x + 3\) (greater than) - Does not match.
• Option #4: \(y \geq \frac{3}{2}x + 3\) (greater than or equal to) - Matches the shaded region, considering that the shading indicates solutions to the right.

Thus, the correct answer is: Option #4.