The lines represent the inequalities [tex]\boxed{y \leqslant \frac{1}{2}x + 2}[/tex]. Option (a) is correct.
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.[tex]y \leqslant \dfrac{1}{2}x + 2[/tex]
b.[tex]y \geqslant \dfrac{1}{2}x + 2[/tex]
c.[tex]y \leqslant \dfrac{1}{3}x + 2[/tex]
d.[tex]y \geqslant \dfrac{1}{3}x + 2[/tex]
Explanation:
The black line intersects y-axis at [tex]\left( {0,2}\right)[/tex], therefore the y-intercept is 2.
The blue line intersect the points that are [tex]\left( { - 4,0}\right)[/tex] and [tex]\left( {0,2}\right)[/tex].
The slope of the line can be obtained as follows.
[tex]\begin{aligned}m&=\frac{{2 - 0}}{{0 -\left({ - 4}\right)}}\\&= \frac{2}{4}\\&= \frac{1}{2}\\\end{aligned}[/tex]
The slope of the line is [tex]m = \dfrac{1}{2}[/tex].
Now check whether the inequality included origin or not.
Substitute [tex]\left( {0,0}\right)[/tex] in equation [tex]y \leqslant\dfrac{1}{2}x + 2[/tex].
[tex]\begin{aligned}&0\leqslant\frac{1}{2}\left(0\right)+ 2\\&0\leqslant 2\\\end{aligned}[/tex]
0 is less than 2 which means that the inequality does include origin.
Therefore, the lines represent the inequalities [tex]\boxed{y\leqslant\dfrac{1}{2}x + 2}[/tex]. Option (a) is correct.
Option (b) is not correct as the equation of the line is [tex]\boxed{y \leqslant\frac{1}{2}x + 2}[/tex].
Option (c) is not correct as the equation of the line is [tex]\boxed{y \leqslant\frac{1}{2}x + 2}[/tex].
Option (d) is not correct as the equation of the line is [tex]\boxed{y \leqslant\frac{1}{2}x + 2}[/tex].
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 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.
