You haven't provided the graph, therefore, I can only help with the concept.
First, you need to define the line drawn:
Equation of line is: y=mx + c
where:
m is the slope = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
c is the y-intercept which is the value of the y at x = 0
The, you need define the shaded region:
To do so, take any point from the graph and substitute in the inequality, if it satisfies the inequality, then, it is within the shaded region, otherwise, it is not.
Hope this helps :)
The line represents the inequality [tex]\boxed{y < - 2x + 3}.[/tex] Option (d) is correct.
Further explanation:
The linear equation with slope [tex]m[/tex] and intercept [tex]c[/tex] 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 > 2x + 3[/tex]
(b). [tex]y < 2x + 3[/tex]
(c). [tex]y > - 2x + 3[/tex]
(d). [tex]y < - 2x + 3[/tex]
Explanation:
The line intersects y-axis at [tex]\left( {0,3} \right)[/tex], therefore the y-intercept is 3.
The points are [tex]\left( {0,3} \right)[/tex] and [tex]\left( {1,1} \right).[/tex]
The slope of the line can be obtained as follows.
[tex]\begin{aligned}m&=\frac{{1 - 3}}{{1 - 0}}\\&= \frac{{ - 2}}{1}\\&=- 2\\\end{aligned}[/tex]
The slope of the line is [tex]m = 2.[/tex]
Now check whether the inequality included origin or not.
Substitute [tex]\left( {0,0} \right)[/tex] in the inequality [tex]y < -2x+3.[/tex]
[tex]\begin{aligned}0&< - 2\left(0\right)+ 3\\0&< 3\\\end{aligned}[/tex]
0 is less than 3 which means that the inequality does include origin.
Kindly refer to the image attached.
The line represents the inequality [tex]\boxed{y < - 2x + 3}.[/tex] Option (d) is correct.
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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.
