Question

Which linear inequality is represented by the graph?
y < 3x + 2
y > 3x + 2
y < x + 2
y > x + 2

Answer 1

Answer: The graph is represented by the inequality y>3x+2

Step-by-step explanation:

Linear inequality is like linear equation just we have inequality signs (<,>,≤,≥) instead of equal sign(=).

In the given question we have the linear inequality represented by graph

We can see the line in the graph passing through the point (-3,-7)

Let's check which option satisfy the point

1) y<3x+2 for this the line should be y=3x+2

⇒ -7=3(-3)+2

⇒-7= -9+2⇒-7=-7,which is true.

2) y>3x+2 which similar as first.

3) y<x+2 for this the line should be y=x+2

⇒ -7= -3+2

⇒-7=-1 which is not true.

4) y > x + 2 is similar to third.

Option 3) and 4) cannot be the required linear inequality.

Now from 1) and 2) , 2) should be the required linear inequality as the graph is shaded above the line and that must be for y (>) greater than inequality. [ for y (<) less than inequality the graph must be shaded below the line]

Therefore, 2)  y > 3x + 2 is the required linear inequality which is represented by the graph.

Answer 2

The linear equality represented by the graph is $\boxed{{\mathbf{y>3x+2}}}$ and it matches with $\boxed{{\mathbf{OPTION B}}}$ .

Further explanation:

It is given that a line passes through points $\left({0,2}\right)$ and $\left({-3,-7}\right)$  as shown below in Figure 1.

The slope of a line passes through points $\left({{x_1},{y_1}}\right)$ and $\left( {{x_2},{y_2}}\right)$is calculated as follows:

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

Here, the slope of a line is denoted as $m$ and points are $\left({{x_1},{y_1}}\right)$ and $\left({{x_2},{y_2}}\right)$.

Substitute $0$  for ${x_1}$ , $2$  for ${y_1}$ , $-3$  for ${x_2}$  and $-7$  for ${y_2}$  in equation (1) to obtain the slope of a line that passes through points $\left({0,2}\right)$ and $\left({-3,-7}\right)$ .

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

Therefore, the slope is $3$ .

The point-slope form of the equation of a line with slope $m$  passes through point $\left({{x_1},{y_1}}\right)$ is represented as follows:

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

Substitute $0$  for ${x_1}$ ,  $2$ for ${y_1}$  and $3$  for $m$  in equation (2) to obtain the equation of line.

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

Therefore, the value of $y$  is $3x+2$ .

Since the shaded part in Figure 1 is above the equation of line $y=3x+2$, therefore, greater than sign is used instead of is equal to.

Thus, the linear inequality is $y>3x+2$  as shown below in Figure 2.

Now, the four options are given below.

$\begin{aligned}{\text{OPTION A}}\to &y<3x+2\hfill\\{\text{OPTION B}}\to &y>3x+2\hfill\\{\text{OPTION C}}\to &y<x+2\hfill\\{\text{OPTION D}}\to &y>x+2\hfill\\\end{aligned}$

Since OPTION B matches the obtained equation that is $y>3x+2$ .

Thus, the linear equality represented by the graph is $\boxed{{\mathbf{y>3x+2}}}$  and it matches with $\boxed{{\mathbf{OPTION B}}}$.

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2. What is the value of $x$ in the equation $x-y=30$  when $y=15$ ? https://brainly.com/question/3965451

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Answer 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