1)x>-2
y<=1
y>=0.5x-2
y<=-2x+3
2)x>=0
y>=0
y<=4x-4
y<=x+5
3)x<=0
y>=0
y<=9
y>=-2x-7
3)x<0
x<=5
y>=1/5x-3
y<=-x+4
4)(7 5)
(8 6)
To solve each system of inequalities, we will first graph them on a coordinate plane.
For easier visualization, let's graph each inequality individually and shade the region that satisfies the inequality.
1) x > -2
To graph this inequality, draw a vertical line passing through x = -2 and shade the region to the right of the line.
2) y ≤ 1
To graph this inequality, draw a horizontal line passing through y = 1 and shade the region below the line.
3) y ≥ 0.5x - 2
To graph this inequality, rearrange it to slope-intercept form (y = mx + b): y ≥ 0.5x - 2.
Start by plotting the y-intercept at y = -2. Since it is a "greater than or equal to" inequality, we'll use a solid line and shade the region above the line.
The slope 0.5 means moving one unit to the right (x+1) results in moving 0.5 units up (y+0.5), so plot another point (1, -1.5) and draw a straight line passing through the two points.
4) y ≤ -2x + 3
To graph this inequality, rearrange it to slope-intercept form (y = mx + b): y ≤ -2x + 3.
Start by plotting the y-intercept at y = 3. Since it is a "less than or equal to" inequality, we'll use a solid line and shade the region below the line.
The slope -2 means moving one unit to the right (x+1) results in moving 2 units down (y-2), so plot another point (1, 1) and draw a straight line passing through the two points.
The solution to this system of inequalities is the region that satisfies all four inequalities. Therefore, it is the overlapping shaded region on the graph that satisfies all the conditions.
Similarly, repeat the above steps for the second, third, and fourth systems of inequalities.
2) x ≥ 0
Draw a vertical line passing through x = 0 and shade the region to the right of the line.
y ≥ 0
Draw a horizontal line passing through y = 0 and shade the region above the line.
y ≤ 4x - 4
Rearrange it to slope-intercept form: y ≤ 4x - 4.
The y-intercept is at (0, -4). Use a solid line and shade the region below the line.
y ≤ x + 5
Rearrange it to slope-intercept form: y ≤ x + 5.
The y-intercept is at (0, 5). Use a solid line and shade the region below the line.
The solution is the overlapping shaded region that satisfies all four inequalities.
3) x ≤ 0
Draw a vertical line passing through x = 0 and shade the region to the left of the line.
y ≥ 0
Draw a horizontal line passing through y = 0 and shade the region above the line.
y ≤ 9
Draw a horizontal line passing through y = 9 and shade the region below the line.
y ≥ -2x - 7
Rearrange it to slope-intercept form: y ≥ -2x - 7.
The y-intercept is at (0, -7). Use a solid line and shade the region above the line.
The solution is the overlapping shaded region that satisfies all four inequalities.
4) x < 0
Draw a vertical line passing through x = 0 and shade the region to the left of the line.
x ≤ 5
Draw a vertical line passing through x = 5 and shade the region to the left of the line.
y ≥ 1/5x - 3
Rearrange it to slope-intercept form: y ≥ (1/5)x - 3.
The y-intercept is at (0, -3). Use a solid line and shade the region above the line.
y ≤ -x + 4
Rearrange it to slope-intercept form: y ≤ -x + 4.
The y-intercept is at (0, 4). Use a solid line and shade the region below the line.
The solution is the overlapping shaded region that satisfies all four inequalities.
For (7, 5) and (8, 6), plot these points on the coordinate plane and see if they fall within the shaded region for each system of inequalities. If both points fall within the shaded region, they satisfy the system of inequalities.