Question

Which number completes the system of linear inequalities represented by the graph?

y >= 2x – 2 and x + 4y >= ?

Answer 1

we have

$y \geq2x-2$ ------> inequality A

The inequality A is the solid red line in the graph

the solution of the inequality A is the shaded area above the solid red line

$x+4y \geq ?$ ------> inequality B

The inequality B is the solid blue line in the graph

the solution of the inequality B is the shaded area above the solid blue line

we know that

the point $(0,-3)$ lie on the line of the inequality B

substitute the values of x and y in the equation of the line

$x+4y = c$

$0+4*(-3) =c$

$c=-12$

so

the equation of the line B is

$x+4y =-12$

the equation of the inequality B is

$x+4y \geq -12$

therefore

the answer is

The number is $-12$

Answer 2

The number is $\boxed{-12}$ in the inequality $\boxed{x+4y\geqslant -12}$.

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,

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

Explanation:

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

The orange line intersect the points that are $\left({1,0}\right)$ and $\left({0,-2}\right)$.

The slope of the line can be obtained as follows.

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

The slope of the line is m = 2.

Therefore, the orange line is $y\geqslant 2x-2$.

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 $\left({-4,-2}\right)$ and $\left({0,-3}\right)$.

The slope of the line can be obtained as follows.

$\begin{aligned}m&=\frac{{-3-\left({-2}\right)}}{{0-\left({-4}\right)}}\\&=\frac{{-3+2}}{4}\\&=-\frac{1}{4}\\\end{aligned}$

The slope of the line is $m=-\frac{1}{4}$.

The inequalities is $x+4y\geqslant b$ passes through the point $\left({0,-3}\right)$.

$\begin{aligned}\left(0\right)+4\left({-3}\right)&=b\\-12&=b\\\end{aligned}$

The number is – 12.

The number is $\boxed{-12}$ in the inequality $\boxed{x+4y\geqslant -12}$.

Learn more:

1. Learn more about inverse of the function https://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 https://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.