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1 month ago

Which linear inequality is represented by the graph?

Mathematics

2 answers:

zzz [12.3K]1 month ago

6 0

The graph illustrates the linear equality represented by $\boxed{{\mathbf{y>3x+2}}}$ , which corresponds to $\boxed{{\mathbf{OPTION B}}}$ .

Additional details:

A line is shown to pass through the points $\left({0,2}\right)$ and $\left({-3,-7}\right)$ in Figure 1 below.

The slope of the line connecting points $\left({{x_1},{y_1}}\right)$ and $\left( {{x_2},{y_2}}\right)$ can be computed as follows:

$m=\frac{{{y_2}-{y_1}}}{{{x_2}-{x_1}}}$ ...........(1)

In this case, the slope is represented by $m$ , while the points are $\left({{x_1},{y_1}}\right)$ and $\left({{x_2},{y_2}}\right)$ .

Replace $0$ with ${x_1}$ , $2$ with ${y_1}$ , $-3$ with ${x_2}$ , and $-7$ with ${y_2}$ in equation (1) to find the slope for the line through points $\left({0,2}\right)$ and $\left({-3,-7}\right)$ .

$\begin{aligned}m&=\frac{{-7-2}}{{-3-0}}\\&=\frac{{-9}}{{-3}}\\&=3\\\end{aligned}$

Consequently, the slope is $3$ .

The equation for a line in point-slope form with slope $m$ that passes through $\left({{x_1},{y_1}}\right)$ is given as follows:

$y-{y_1}=m\left({x-{x_1}}\right)$ ...........(2)

By substituting $0$ for ${x_1}$ , $2$ for ${y_1}$ , and $3$ for $m$ in equation (2), the equation of the line can be derived.

$\begin{aligned}y-2&=3\left({x-0}\right)\\y-2&=3x\\y&=3x+2\\\end{aligned}$

Thus, the value for $y$ turns out to be $3x+2$ .

Noting that the shaded area in Figure 1 exists above the line represented by $y=3x+2$ , a greater than sign is employed instead of equality.

Therefore, the linear inequality depicted is $y>3x+2$ , as seen in Figure 2 below.

Consequently, four options are provided below.

$\begin{aligned}{\text{OPTION A}}\to &y3x+2\hfill\\{\text{OPTION C}}\to &yx+2\hfill\\\end{aligned}$

Since OPTION B corresponds to the derived equation $y>3x+2$ .

Thus, the graph shows the linear equality of $\boxed{{\mathbf{y>3x+2}}}$ , aligning with $\boxed{{\mathbf{OPTION B}}}$ .

To explore further:

1. Which classification best describes this system of equations?

2. What is the value of $x$ in the equation $x-y=30$ when $y=15$ ?

3. What are the values of x?

Response Details:

Grade: Junior High School

Subject: Mathematics

Chapter: Coordinate Geometry

Keywords: Coordinate Geometry, linear equation, system of linear equations in two variables, variables, mathematics, inequality

PIT_PIT [12.4K]1 month ago

5 0

Answer: The graph corresponds to the inequality y > 3x + 2

Step-by-step explanation:

A linear inequality is similar to a linear equation, but it consists of inequality symbols (<, >, ≤, ≥) rather than the equal sign (=).

In the question, the linear inequality is depicted through a graph.

The line in the graph intersects the point (-3,-7).

Now, let’s evaluate which option meets the conditions of this point.

1) y < 3x + 2 implies the line should be y = 3x + 2.

⇒ -7 = 3(-3) + 2.

⇒ -7 = -9 + 2 ⇒ -7 = -7, which holds true.

2) y > 3x + 2, which is analogous to the first.

3) y < x + 2 means the line should be y = x + 2.

⇒ -7 = -3 + 2.

⇒ -7 = -1, which is false.

4) y > x + 2 parallels the third scenario.

Thus, options 3) and 4) cannot represent the required linear inequality.

From options 1) and 2), option 2) is the correct linear inequality since the graph indicates shading above the line, suggesting y (>) must be greater. [Conversely, for y (<) to be lower, the shading would be below the line.]

Consequently, the correct linear inequality presented by the graph is 2) y > 3x + 2.

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