Question

Which linear inequality is represented by the graph?

Answer 1

The linear equality represented by the graphis $\boxed{{\mathbf{y > 3x + 2}}}$ 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:

$m=\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$       ......(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 $\boxed{{\mathbf{OPTION B}}}$.

Learn more:

1. Which classification best describes the following system of equations? brainly.com/question/9045597

2. What is the value of   in the equation  when  ? brainly.com/question/3965451

3. What are the values of x?brainly.com/question/2093003

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,equation of line, line, passes through point, inequality

Answer 2

Answer-

$\boxed{\boxed{y > 3x + 2}}$

Solution-

From the graph, the coordinates of two present on the line are, (0, 2), (-3, -7)

Applying two point formula for straight line,

$\Rightarrow \dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}$

Here,

x₁ = 0

y₁ = 2

x₂ = -3

y₂ = -7

Putting in the formula,

$\Rightarrow \dfrac{y-2}{-7-2}=\dfrac{x-0}{-3-0}$

$\Rightarrow \dfrac{y-2}{-9}=\dfrac{x}{-3}$

$\Rightarrow \dfrac{y-2}{3}=\dfrac{x}{1}$

$\Rightarrow \dfrac{y-2}{3}=x$

$\Rightarrow {y-2}=3x$

$\Rightarrow y=3x+2$

As we have got the equality equation, now we have to care about the sign.

So option 3 and 4 are not the correct options.

Putting origin (0, 0) in the first equation i.e $y < 3x + 2$,

$\Rightarrow 0$

As this satisfies the equation the shaded region will be towards origin or right side of the line $y=3x+2$. This area is the opposite to the desired area, so this is not the right answer.

Putting origin (0, 0) in the second equation i.e $y > 3x + 2$,

$\Rightarrow 0>3(0)+2\\\\\Rightarrow 0\ngtr 2$

As this does not satisfy the equation the shaded region will be opposite to origin or left side of the line $y=3x+2$, so this is the right answer.