y < 2/3x + 3
Explanation:
In the graph, the grey region corresponds to the region non-allowed by the inequality.
We see that for x=0, y is allowed to be only less than 3: this means that the correct inequality must be in the form y<mx+3, so only the 1st option or the 4th option.
In order to choose the correct option, we should find the value of m, the slope of the line in the graph. This slope can be found by calculating the variation of y divided by the variation of x:
[tex]m=\frac{\Delta y}{\Delta x}[/tex]
Choosing for example the points x=0 (which corresponds to y=3) and x=3 (which corresponds to y=5), we find
[tex]m=\frac{5-3}{3-0}=\frac{2}{3}[/tex]
So, the equation of the line is
[tex]y=\frac{2}{3}x+3[/tex]
and so the correct inequality is
[tex]y<\frac{2}{3}x+3[/tex]
The linear equality represented by the graphis [tex]\boxed{{\mathbf{y > }}\frac{{\mathbf{2}}}{{\mathbf{3}}}{\mathbf{x + 3}}}[/tex] and it matches with [tex]\boxed{{\mathbf{OPTION C}}}[/tex].
Further explanation:
From given graph, theline passes through points [tex]\left({3,5}\right)[/tex] and [tex]\left({ - 3,1}\right)[/tex] as shown below in Figure 1.
The slope of a line passes through points [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex] is calculated as follows:
[tex]m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}{\text{}}[/tex] ......(1)
Here, the slope of a line is denoted as and points are [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex].
Substitute [tex]3[/tex] for [tex]{x_1}[/tex] , [tex]5[/tex] for [tex]{y_1}[/tex] , [tex]-3[/tex] for [tex]{x_2}[/tex] and [tex]1[/tex] for [tex]{y_2}[/tex] in equation (1) to obtain the slope of a line that passes through points [tex]\left( {3,5}\right)[/tex] and [tex]\left({ - 3,1}\right)[/tex].
[tex]\begin{aligned}m&=\frac{{1 - 5}}{{ - 3 - 3}}\\&=\frac{{ - 4}}{{ - 6}}\\&=\frac{2}{3}\\\end{aligned}[/tex]
Therefore, the slope is [tex]\frac{2}{3}[/tex].
The point-slope form of the equation of a line with slope passes through point is represented as follows:
[tex]y - {y_1} = m\left({x - {x_1}}\right){\text{}}[/tex] ......(2)
Substitute [tex]3[/tex] for [tex]{x_1}[/tex] , [tex]5[/tex] for [tex]{y_1}[/tex] and [tex]\frac{2}{3}[/tex] for [tex]m[/tex] in equation (2) to obtain the equation of line.
[tex]\begin{aligned}y - 5 = \frac{2}{3}\left({x - 3} \right)\\3\left( {y - 5} \right) = 2x - 6\\3y - 15 = 2x - 6\\3y = 2x - 6 + 15\\\end{aligned}[/tex]
Further simplify the above equation.
[tex]\begin{aligned}3y=2x + 9\\y=\frac{2}{3}x+\frac{9}{3}\\y=\frac{2}{3}x + 3\\\end{aligned}[/tex]
Therefore, the value of [tex]y[/tex] is [tex]\dfrac{2}{3}x + 3[/tex].
Since the shaded part in Figure 1 is above the equation of line [tex]y = \dfrac{2}{3}x + 3[/tex], therefore, greater than sign is used instead of is equal to
Therefore, the inequality is [tex]y > \dfrac{2}{3}x + 3[/tex].
Now, the four options are given below.
[tex]\begin{aligned}{\text{OPTION A}}\to y < \frac{2}{3}x + 3 \hfill \\{\text{OPTION B}} \to y > \frac{3}{2}x + 3 \hfill \\{\text{OPTION C}} \to y > \frac{2}{3}x + 3 \hfill \\{\text{OPTION D}}\to y < \frac{3}{2}x + 3 \hfill\\\end{aligned}[/tex]
Since OPTION C matches the obtained equation that is [tex]y > \dfrac{2}{3}x + 3[/tex].
Thus, the linear equality represented by the graph is [tex]\boxed{{\mathbf{y > }}\frac{{\mathbf{2}}}{{\mathbf{3}}}{\mathbf{x + 3}}}[/tex]and it matches with [tex]\boxed{{\mathbf{OPTION C}}}[/tex]
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Answer Details:
Grade: Junior High School
Subject: Mathematics
Chapter: Coordinate Geometry
Keywords: Coordinate Geometry, linear equation, system of linear equations in two variables, variables, mathematics,equation of line, line, passes through point, inequality
