Question

Which linear inequality is represented by the graph?
y > 2x + 2
y ≥ x + 1
y > 2x + 1
y ≥ x + 2

Answer 1

Step 1

Find the slope of the line

Let

$A(0,1)\\B(2,5)$

the slope is equal to

$m=\frac{y2-y1}{x2-x1}$

substitute

$m=\frac{5-1}{2-0}$

$m=\frac{4}{2}$

$m=2$

Find the equation of the line

we know that

the equation of the line into slope-intercept form is

$y=mx+b$

where

m is the slope

b is the y-intercept

in this problem we have

$m=2$

$b=1$ ------> the y-intercept is the point A

substitute

$y=2x+1$

Step 2

Find the equation of the inequality

we know that

the solution is the shaded area above the dotted line

so

above dotted line---------> represent the symbol ($>$)

the inequality is

$y>2x+1$

therefore

the answer is

$y>2x+1$

Answer 2

The correct option is $\boxed{\bf option (c)}$ i.e., $\boxed{y>2x+1}$.

Further explanation:

The linear equation of the line is $y=mx+b$ where, $m$ is the slope of the line and $c$ is the $y$-intercept of the line.

Suppose the line passes through the two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$.

Therefore, the slope of the line can be calculated as follows:

$\boxed{m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}}$

The symbol $>$ represents the solution set lies above the dotted line and the symbol $<$ represents the solution set below the dotted line.

Given:

The linear inequalities are  given as follows:

$\boxed{\begin{aligned}y&>2x+2\\ y&\geq x+1\\ y&>2x+1\\ y&\geq x+2\end{aligned}}$

Calculation:

First we will find the equation of the line.

The line cuts the $y$-axis on the coordinate $(0,1)$ as shown in the Figure 1 (attached in the end).

Therefore, the $y$-intercept is $1$.

Kindly refer the Figure attached to the question.

From the figure 1 (attached in the end) we can see that the line passes through the points $(-2,-3)\text{ and }(1,3)$.

The slope of the line can be calculated as follows:

$\begin{aligned}m&=\dfrac{3-(-3)}{1-(-2)}\\&=\dfrac{6}{3}\\&=2\end{aligned}$    

Therefore, the value of $m$ is $m=2$ and the value of $b$ is $b=1$.

Substitute $2$ for $m$ and $1$ for $b$ in the equation of the line $y=mx+b$.

Second we will find the equation of the inequality.

The shaded region is shown in Figure 1 above the dotted line .  

Therefore, the solution set of the line lie above the dotted line and it will represented by the symbol $>$.

Thus, the inequality $y>2x+1$ satisfies the given graph.

Therefore, the correct option is $\boxed{\bf option (c)}$ i.e., $\boxed{y>2x+1}$.

Learn more:

1. Learn more about the representation of the graph https://brainly.com/question/2491745

2. Learn more about the graph of the quadratic function https://brainly.com/question/2334270

3. Learn more about graphed below of the function  https://brainly.com/question/9590016

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Linear inequalities

Keywords:  Linear equations, linear inequality, equation, line, slope, intercept, dotted line , coordinate, shaded region, solutions set, graph, curve.