graph the following system of inequalities on the coordinate plane and shade the resulting solution region…

graph the following system of inequalities on the coordinate plane and shade the resulting solution region: \\(\\begin{cases} y > 2x - 3 \\\\ y \\leq -x + 4 \\end{cases}\\)

graph the following system of inequalities on the coordinate plane and shade the resulting solution region: \\(\\begin{cases} y > 2x - 3 \\\\ y \\leq -x + 4 \\end{cases}\\)

Answer

Explanation:

Step1: Graph ( y > 2x - 3 )

First, graph the line ( y = 2x - 3 ). The slope of this line is ( 2 ) and the y - intercept is ( - 3 ). Since the inequality is ( y>2x - 3 ), we use a dashed line (because the inequality is strict, ( y) is not equal to ( 2x - 3 )) to represent the boundary. Then, to determine which side of the line to shade, we can use a test point. A common test point is ( (0,0) ). Substitute ( x = 0 ) and ( y = 0 ) into the inequality ( y>2x - 3 ): ( 0>2(0)-3), which simplifies to ( 0 > - 3 ). This is true, so we shade the region above the dashed line ( y = 2x - 3 ).

Step2: Graph ( y\leq - x + 4 )

Next, graph the line ( y=-x + 4 ). The slope of this line is ( - 1 ) and the y - intercept is ( 4 ). Since the inequality is ( y\leq - x + 4 ), we use a solid line (because the inequality is non - strict, ( y) can be equal to ( - x + 4 )) to represent the boundary. Again, use a test point, say ( (0,0) ). Substitute into the inequality: ( 0\leq-(0)+4 ), which simplifies to ( 0\leq4 ). This is true, so we shade the region below (or on) the solid line ( y=-x + 4 ).

Step3: Find the solution region

The solution region of the system of inequalities is the region that is shaded for both inequalities simultaneously. It is the region that is above the dashed line ( y = 2x - 3 ) and below (or on) the solid line ( y=-x + 4 ). To find the intersection point of the two lines ( y = 2x-3 ) and ( y=-x + 4 ), set them equal to each other: ( 2x-3=-x + 4 ) Add ( x ) to both sides: ( 3x-3 = 4 ) Add ( 3 ) to both sides: ( 3x=7 ) Divide both sides by ( 3 ): ( x=\frac{7}{3}\approx2.33 ) Substitute ( x = \frac{7}{3} ) into ( y=-x + 4 ): ( y=-\frac{7}{3}+4=\frac{- 7 + 12}{3}=\frac{5}{3}\approx1.67 ) So the two lines intersect at the point ( (\frac{7}{3},\frac{5}{3}) )

Answer:

To graph the system:

  1. Draw a dashed line ( y = 2x-3 ) and shade above it.
  2. Draw a solid line ( y=-x + 4 ) and shade below (and on) it.
  3. The solution region is the overlapping shaded area, bounded by the dashed line ( y = 2x - 3 ) (not including the line), the solid line ( y=-x + 4 ) (including the line), and their intersection point ( (\frac{7}{3},\frac{5}{3}) )