The lines represent the inequalities [tex]\boxed{y<2x+4}{\text{ and }}\boxed{x+5y>5}[/tex].
Further explanation:
The linear equation with slope m and intercept c is given as follows.
[tex]\boxed{y=mx+c}[/tex]
The formula for slope of line with points [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex] can be expressed as,
[tex]\boxed{m=\frac{{{y_2}-{y_1}}}{{{x_2} - {x_1}}}}[/tex]
Given:
The inequalities are as follows.
a. x – 5y < –5 and y < –2x – 4
b. x + 5y > 5 and y < 2x + 4
c. x – 5y < –5 and y < –2x + 4
d. x + 5y > 5 and y < 2x – 4
Explanation:
The blue line intersects y-axis at [tex]\left({0,4}\right)[/tex], therefore the y-intercept is 4.
The blue line intersect the points that are [tex]\left({-2,0}\right)[/tex] and [tex]\left({0,4}\right)[/tex].
The slope of the line can be obtained as follows.
[tex]\begin{aligned}m&=\frac{{4-0}}{{0-\left({-2}\right)}}\\&=\frac{4}{2}\\&=2\\\end{aligned}[/tex]
The slope of the line is m = 2.
Now check whether the inequality included origin or not.
Substitute [tex]\left( {0,0} \right)[/tex] in equation y=x+1.
[tex]\begin{aligned}0&>2\left(0\right)+4\hfill\\0&>4\hfill\\\end{aligned}[/tex]
0 is not greater than 1 which means that the inequality doesn’t include origin.
Therefore, the blue line is y < 2x + 4.
The orange line intersects y-axis at [tex]\left({0,1}\right)[/tex], therefore the y-intercept is 1.
The orange line intersect the points that are [tex]\left({5,0}\right)[/tex] and [tex]\left({0,1}\right)[/tex] .
The slope of the line can be obtained as follows.
[tex]\begin{aligned}m&=\frac{{1-0}}{{0-\left(5\right)}}\\&=\frac{1}{{-5}}\\&=-\frac{1}{5}\\\end{aligned}[/tex]
The slope of the line is m = [tex]- \frac{1}{5}[/tex].
Now check whether the inequality included origin or not.
Substitute [tex]\left({0,0}\right)[/tex] in equation y > [tex]-\frac{1}{5}x+1[/tex].
[tex]\begin{aligned}0&>-\frac{1}{5}\left(0\right)+1\hfill\\0&>1\hfill\\\end{aligned}[/tex]
0 is not greater than 1 which means that the inequality doesn’t include origin.
Therefore, the orange line is y > [tex]-\frac{1}{5}x+1[/tex].
Solve the equation y > [tex]-\frac{1}{5}x+1[/tex].
[tex]\begin{aligned}y&>-\frac{1}{5}x+1\\5y&>-x+5\\5y+x&>5\\\end{aligned}[/tex]
The equation of orange line is 5y + x > 5.
Option (a) is not correct as it satisfy the inequalities of the graph.
Option (b) is correct as it satisfy the inequalities of the graph.
Option (c) is not correct as it satisfy the inequalities of the graph.
Option (d) is is not correct as it satisfy the inequalities of the graph.
Hence, [tex]\boxed{{\text{Option (b)}}}[/tex] is correct.
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.
