Question

graph the solution to the following system of inequalities.
$y > -3x + 5$
$y < 4x - 5$

Explanation:

Step1: Graph \( y > -3x + 5 \)

First, graph the line \( y = -3x + 5 \). The slope is \(-3\) and the y - intercept is \( 5 \). Since the inequality is \( y > -3x + 5 \), the line should be dashed (because the inequality is strict, \( y\) is not equal to \(-3x + 5\)). Then, shade the region above the line (because for a point \((x,y)\) to satisfy \( y > -3x + 5\), \( y\) must be greater than the value of \(-3x + 5\) at a given \( x\)).

Step2: Graph \( y < 4x - 5 \)

Next, graph the line \( y = 4x - 5 \). The slope is \( 4 \) and the y - intercept is \(-5\). Since the inequality is \( y < 4x - 5 \), the line should be dashed (because \( y\) is not equal to \( 4x - 5\)). Then, shade the region below the line (because for a point \((x,y)\) to satisfy \( y < 4x - 5\), \( y\) must be less than the value of \( 4x - 5\) at a given \( x\)).

Step3: Find the Intersection Region

The solution to the system of inequalities is the region where the two shaded regions overlap. To find the intersection of the two lines \( y=-3x + 5\) and \( y = 4x-5\), set \(-3x + 5=4x - 5\). Solving for \( x\):

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Substitute \( x = \frac{10}{7}\) into \( y=-3x + 5\): \( y=-3\times\frac{10}{7}+5=\frac{-30 + 35}{7}=\frac{5}{7}\approx0.71\). The intersection point of the two lines is \((\frac{10}{7},\frac{5}{7})\). The overlapping region (the solution) is the area that is above \( y=-3x + 5\) (dashed line) and below \( y = 4x - 5\) (dashed line), and to the right of the intersection point (since for \( x>\frac{10}{7}\), \( 4x-5>-3x + 5\)).

Answer:

The graph should have two dashed lines: \( y=-3x + 5\) (slope - 3, y - intercept 5, shaded above) and \( y = 4x-5\) (slope 4, y - intercept - 5, shaded below), with the overlapping region (the solution) being the area that satisfies both inequalities. The intersection point of the two lines is \((\frac{10}{7},\frac{5}{7})\). (Note: Since the question asks to graph the solution, the final answer is the description of the graph as above, with the two dashed lines and the overlapping shaded region.)