Question

Which linear inequality is represented by the graph

Answer 1

The lines represent the inequalities $\boxed{y \leqslant \frac{1}{2}x + 2}$. Option (a) is correct.

Further explanation:

The linear equation with slope m and intercept c is given as follows.

$\boxed{y = mx + c}$

The formula for slope of line with points $\left( {{x_1},{y_1}}\right)$ and $\left( {{x_2},{y_2}}\right)$ can be expressed as,

$\boxed{m=\frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}}$

Given:

The inequalities are as follows.

a.$y \leqslant \dfrac{1}{2}x + 2$  

b.$y \geqslant \dfrac{1}{2}x + 2$  

c.$y \leqslant \dfrac{1}{3}x + 2$  

d.$y \geqslant \dfrac{1}{3}x + 2$

Explanation:

The black line intersects y-axis at $\left( {0,2}\right)$, therefore the y-intercept is 2.

The blue line intersect the points that are $\left( { - 4,0}\right)$ and  $\left( {0,2}\right)$.

The slope of the line can be obtained as follows.

$\begin{aligned}m&=\frac{{2 - 0}}{{0 -\left({ - 4}\right)}}\\&= \frac{2}{4}\\&= \frac{1}{2}\\\end{aligned}$

The slope of the line is $m = \dfrac{1}{2}$.

Now check whether the inequality included origin or not.

Substitute $\left( {0,0}\right)$ in equation $y \leqslant\dfrac{1}{2}x + 2$.

$\begin{aligned}&0\leqslant\frac{1}{2}\left(0\right)+ 2\\&0\leqslant 2\\\end{aligned}$

0 is less than 2 which means that the inequality does include origin.

Therefore, the lines represent the inequalities $\boxed{y\leqslant\dfrac{1}{2}x + 2}$. Option (a) is correct.

Option (b) is not correct as the equation of the line is $\boxed{y \leqslant\frac{1}{2}x + 2}$.

Option (c) is not correct as the equation of the line is $\boxed{y \leqslant\frac{1}{2}x + 2}$.

Option (d) is not correct as the equation of the line is  $\boxed{y \leqslant\frac{1}{2}x + 2}$.

 

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear inequalities

Keywords: numbers, slope, slope intercept, inequality, equation, linear inequality, shaded region, y-intercept, graph, representation, origin.

Answer 2

The answer would be A. First you have to look at the slope, which is 1/2, so it can't be C or D. Then look at where the shaded region is. The shaded region is below the line, so the y value would be less than or equal to the expression.