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

A dotted line drawn on a coordinate plane slopes downward from top left to bottom right. The part of the plane above and to the right of the line is shaded. Both axes range from negative 10 to 10 in unit increments.

Write an equation for the linear inequality in the provided graph. Which option is the correct equation for the graph?

Option #1: y≥−x−2


Option #2: y>−x−2


Option #3: y≤−x−2


Option #4: y<−x−2


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

11 answers

To find the correct equation for the linear inequality represented by the graph, we analyze the following:

The dotted line suggests that the inequality is strict, meaning it does not include points on the line itself. This indicates we will use either ">" or "<" in our inequality.

The line slopes downward, which typically corresponds to a negative slope in the form of \(y = mx + b\). Since the line appears to have a negative slope of -1 (based on its downward trajectory from left to right), we can represent it as \(y = -x + b\).

Looking at the equation \(y = -x - 2\), we can determine the specific boundary line mentioned. The inequality must include all points above this line (the shaded area above).

Given that the line is not inclusive of its own points (due to the dotted line), we can conclude that the correct form of the inequality for the shaded region above the line is \(y > -x - 2\).

Thus, the correct option is:

Option #2: \(y > -x - 2\)

So, the final answer is:

The equation that matches the graph is Option #2.

Which of the following points is a solution to the linear inequality y>1/9x−2/9 ?(1 point)
Responses

(0,−1/9)
left parenthesis 0 comma negative Start Fraction 1 over 9 End Fraction right parenthesis

(0,−4/9)
left parenthesis 0 comma negative Start Fraction 4 over 9 End Fraction right parenthesis

(0,−1/3)
left parenthesis 0 comma negative Start Fraction 1 over 3 End Fraction right parenthesis

(0,−2/9)

To determine which point is a solution to the inequality \( y > \frac{1}{9}x - \frac{2}{9} \), we can substitute the x-coordinate of each point into the equation and check if the corresponding y-value satisfies the inequality.

The inequality can be rewritten in terms of specific x-values. Since all the given points have \( x = 0 \), we can substitute \( x = 0 \) into the inequality:

\[ y > \frac{1}{9}(0) - \frac{2}{9} \]

This simplifies to:

\[ y > -\frac{2}{9} \]

Now, we will check each of the given points:

  1. Point (0, -1/9): \[ -\frac{1}{9} > -\frac{2}{9} \quad \text{(True)} \]

  2. Point (0, -4/9): \[ -\frac{4}{9} > -\frac{2}{9} \quad \text{(False)} \]

  3. Point (0, -1/3): \[ -\frac{1}{3} = -\frac{3}{9} > -\frac{2}{9} \quad \text{(False)} \]

  4. Point (0, -2/9): \[ -\frac{2}{9} > -\frac{2}{9} \quad \text{(False)} \]

The only point that satisfies the inequality \( y > -\frac{2}{9} \) is (0, -1/9).

Therefore, the solution to the linear inequality is:

(0, -1/9).

Use the image to answer the question.

A coordinate plane shows a graph of an upward-sloping, diagonal, dashed line, with shading extending from the line to the left. The line starts in quadrant 3, extends through quadrant 4, and ends in quadrant 1.

Is the point (0,−3) a solution to the inequality y>4/5x−3 , as shown on the graph?

(1 point)
Responses

No, because (0,−3) falls on the boundary line.
No, because left parenthesis 0 comma negative 3 right parenthesis falls on the boundary line.

Yes, because (0,−3) falls on the boundary line.
Yes, because left parenthesis 0 comma negative 3 right parenthesis falls on the boundary line.

No, because (0,−3) falls within the shaded region.
No, because left parenthesis 0 comma negative 3 right parenthesis falls within the shaded region.

Yes, because (0,−3) falls within the shaded region.

To determine whether the point \((0, -3)\) is a solution to the inequality \(y > \frac{4}{5}x - 3\), we need to evaluate it against the boundary line represented by the equation \(y = \frac{4}{5}x - 3\).

  1. Substituting \(x = 0\) into the equation:

    \[ y = \frac{4}{5}(0) - 3 = -3 \]

  2. Comparing \(y\)-values:

    The calculated \(y\)-value when \(x = 0\) is \(-3\).

  3. Analyzing the point:

    The point \((0, -3)\) lies exactly on the boundary line since \(y = -3\).

  4. Understanding the inequality:

    Since the inequality is \(y > \frac{4}{5}x - 3\), for any point on the boundary line, the inequality will not hold because the inequality does not include equality (indicated by the absence of \(\geq\)).

Therefore, since \((0, -3)\) is on the boundary line and the inequality is strictly greater than, we can conclude:

No, because (0, -3) falls on the boundary line.

Show which of the following points is part of the solution set for the inequality y≤2/5x+1 .(1 point)
Responses

(−10,6)
left parenthesis negative 10 comma 6 right parenthesis

(10,6)
left parenthesis 10 comma 6 right parenthesis

(10,4)
left parenthesis 10 comma 4 right parenthesis

(10,5.5)

To determine which of the given points is part of the solution set for the inequality \( y \leq \frac{2}{5}x + 1 \), we can substitute the \( x \)-coordinate of each point into the equation \( y = \frac{2}{5}x + 1 \) and see if the corresponding \( y \)-value satisfies the inequality.

Let's evaluate each point:

  1. Point \((-10, 6)\): \[ y = \frac{2}{5}(-10) + 1 = -4 + 1 = -3 \] Check the inequality: \[ 6 \leq -3 \quad \text{(False)} \]

  2. Point \((10, 6)\): \[ y = \frac{2}{5}(10) + 1 = 4 + 1 = 5 \] Check the inequality: \[ 6 \leq 5 \quad \text{(False)} \]

  3. Point \((10, 4)\): \[ y = \frac{2}{5}(10) + 1 = 4 + 1 = 5 \] Check the inequality: \[ 4 \leq 5 \quad \text{(True)} \]

  4. Point \((10, 5.5)\): \[ y = \frac{2}{5}(10) + 1 = 4 + 1 = 5 \] Check the inequality: \[ 5.5 \leq 5 \quad \text{(False)} \]

After evaluating all the points, we see that the only point that satisfies the inequality \( y \leq \frac{2}{5}x + 1 \) is:

(10, 4).

Write a two-variable linear inequality that models the graph of the solid line y=2x−4 with a shaded region above the line. Which of the following options is correct?(1 point)
Responses

y≤2x−4
y less-than-or-equal-to 2 x minus 4

y≥2x−4
y greater-than-or-equal-to 2 x minus 4

y<2x−4
y less than 2 x minus 4

y>2x−4

To determine the correct two-variable linear inequality that models the graph of the solid line \(y = 2x - 4\) with a shaded region above the line, we need to consider two key aspects:

  1. Nature of the line: Since the line is solid, this indicates that points on the line itself are included in the solution set. Therefore, we will use the "greater than or equal to" or "less than or equal to" form of the inequality.

  2. Shaded region: The shaded region is above the line, which indicates that we are looking for values of \(y\) that are greater than or equal to the expression \(2x - 4\).

Putting these together, we arrive at the inequality:

\[ y \geq 2x - 4 \]

Thus, the correct option is:

y ≥ 2x − 4.

Use the image to answer the question.

A solid line drawn on a coordinate plane slopes steeply upward from bottom left to top right. The part of the plane to the right of the line is shaded. Both axes range from negative 10 to 10 in unit increments.

Which linear inequality matches the given graph?

(1 point)
Responses

y>2x+2
y greater than 2 x plus 2

y≥2x+2
y greater-than-or-equal-to 2 x plus 2

y≤2x+2
y less-than-or-equal-to 2 x plus 2

y<2x+2

To determine the correct linear inequality that matches the description of the graph, we need to consider two important aspects based on the provided information:

  1. Nature of the line: Since the line is described as solid, this indicates that the inequality includes equality. Therefore, we will use either "greater than or equal to" (≥) or "less than or equal to" (≤).

  2. Shaded region: The shaded region is to the right of the line, indicating that we are looking for values of \(y\) that are greater than or equal to the expression represented by the line.

Given that the line is in the form \(y = mx + b\), and based on the image, if we consider the line equation as \(y = 2x + 2\) (since it slopes upward and appears to intercept at \(y = 2\) when \(x = 0\)), we conclude:

  • Because the region to the right of the line is shaded and the line is solid, the correct inequality that matches the graph is:

\[ y \geq 2x + 2 \]

Therefore, the correct option is:

y ≥ 2x + 2.