Question

Which linear inequality is represented by the graph?

Answer 1

The lines represent the inequalities $\boxed{y\leqslant\frac{1}{2}x + 3}.$

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. $y \leqslant 2x + 4.$

b.$y \leqslant x + 3.$

c.$y \geqslant x + 3.$

d. $y \geqslant 2x + 3.$

Explanation:

The blue line intersects y-axis at $\left( {0,3} \right),$ therefore the y-intercept is 3.

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

The slope of the line can be obtained as follows.

$\begin{aligned}m&= \frac{{2 - 3}}{{ - 2 - 0}}\\&=\frac{{ - 1}}{{ - 2}}\\&=\frac{1}{2}\\\end{aligned}$

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 equationy $\leqslant\dfrac{1}{2}x + 3.$

$\begin{aligned}0\leqslant\frac{1}{2}\left(0\right)+3\hfill\\0\leqslant3\hfill \\\end{aligned}$

0 is less than 3 which means that the inequality does include origin.

Therefore, the blue line is $\boxed{y\leqslant\frac{1}{2}x + 3}$

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

Answer

$y\leq \frac{1}{2} x+3$

Explanation

First we are going to find the equation of the solid line passing trough the points (0, 3) and (2, 4).

Using the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m=\frac{4-3}{2-0}$

$m=\frac{1}{2}$

Now we can use the point slope formula to complete the line equation:

$y-y_{1}=m(x-x_{1})$

$y-3=\frac{1}{2} (x-0)$

$y-3=\frac{1}{2}x$

$y=\frac{1}{2}x+3$

Since the shaded region is bellow the line $y=\frac{1}{2}x+3$, the inequality represented in the graph is $y\leq \frac{1}{2} x+3$.