solve the following system of inequalities graphically on the set of axes below. state the coordinates of a…

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set. \n$y \\leq -\\frac{6}{5}x - 4$\n$y \\geq \\frac{2}{5}x + 4$

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set. \n$y \\leq -\\frac{6}{5}x - 4$\n$y \\geq \\frac{2}{5}x + 4$

Answer

Explanation:

Step1: Graph ( y \leq -\frac{6}{5}x - 4 )

The inequality is in slope - intercept form ( y=mx + b ), where ( m =-\frac{6}{5}) and ( b=-4 ). First, plot the y - intercept at ( (0,-4) ). The slope ( m =-\frac{6}{5}) means from the y - intercept, we go down 6 units and right 5 units (or up 6 units and left 5 units) to find another point. Since the inequality is ( y\leq-\frac{6}{5}x - 4 ), we draw a solid line (because the inequality is non - strict) and shade the region below the line.

Step2: Graph ( y \geq \frac{2}{5}x + 4 )

This inequality is also in slope - intercept form with ( m=\frac{2}{5}) and ( b = 4 ). Plot the y - intercept at ( (0,4) ). Using the slope ( \frac{2}{5}), from the y - intercept, we go up 2 units and right 5 units (or down 2 units and left 5 units) to find another point. Since the inequality is ( y\geq\frac{2}{5}x + 4 ), we draw a solid line (non - strict inequality) and shade the region above the line.

Step3: Find the intersection of the two shaded regions

The solution set of the system of inequalities is the region that is shaded by both inequalities. To find a point in the solution set, we can find the intersection point of the two lines ( y =-\frac{6}{5}x - 4 ) and ( y=\frac{2}{5}x + 4 ). Set the two equations equal to each other: [ -\frac{6}{5}x-4=\frac{2}{5}x + 4 ] Add ( \frac{6}{5}x ) to both sides: [ -4=\frac{2}{5}x+\frac{6}{5}x + 4 ] [ -4=\frac{8}{5}x + 4 ] Subtract 4 from both sides: [ -8=\frac{8}{5}x ] Multiply both sides by ( \frac{5}{8} ): [ x=-5 ] Substitute ( x = - 5 ) into ( y=\frac{2}{5}x + 4 ): [ y=\frac{2}{5}(-5)+4=-2 + 4=2 ] So the intersection point of the two lines is ( (-5,2) ). We can also test a point in the overlapping region. Let's test ( (-5,2) ) in both inequalities:

  • For ( y\leq-\frac{6}{5}x - 4 ): Substitute ( x=-5,y = 2 ). ( 2\leq-\frac{6}{5}(-5)-4=6 - 4=2 ), which is true.
  • For ( y\geq\frac{2}{5}x + 4 ): Substitute ( x=-5,y = 2 ). ( 2\geq\frac{2}{5}(-5)+4=-2 + 4=2 ), which is true.

Answer:

A point in the solution set is ((-5,2)) (other valid points can also be found by testing points in the overlapping shaded region).