The lines represent the inequalities [tex]\boxed{y\leqslant\frac{1}{2}x + 3}.[/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. [tex]y \leqslant 2x + 4.[/tex]
b.[tex]y \leqslant x + 3.[/tex]
c.[tex]y \geqslant x + 3.[/tex]
d. [tex]y \geqslant 2x + 3.[/tex]
Explanation:
The blue line intersects y-axis at [tex]\left( {0,3} \right),[/tex] therefore the y-intercept is 3.
The blue line intersect the points that are [tex]\left({-2,2}\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 - 3}}{{ - 2 - 0}}\\&=\frac{{ - 1}}{{ - 2}}\\&=\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 equationy [tex]\leqslant\dfrac{1}{2}x + 3.[/tex]
[tex]\begin{aligned}0\leqslant\frac{1}{2}\left(0\right)+3\hfill\\0\leqslant3\hfill \\\end{aligned}[/tex]
0 is less than 3 which means that the inequality does include origin.
Therefore, the blue line is [tex]\boxed{y\leqslant\frac{1}{2}x + 3}[/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.
