Graph the solution set for the of linear inequalities.
x−y≤4
x+3y≤4
x≥−5
x−y≤4 -----> y ≥ x-4
x+3y≤4 ---> y ≤ (4 - x)/3
x≥−5
plot y = x-4, shade in the region above it
plot y = (4-x)/3 , shade in the regions below it
plot x = -5, a vertical line, and shade in the region to the right of it
To graph the solution set for a system of linear inequalities, you need to follow these steps:
Step 1: Graph each inequality individually on the coordinate plane.
- Start with the first inequality: x - y ≤ 4
- Convert it to the slope-intercept form (y = mx + b) by isolating y:
y ≥ x - 4
- To graph this inequality, draw a dashed line with a slope of 1 (since the coefficient of x is 1) and a y-intercept of -4. Make the line dashed because the inequality is greater than or equal to (≥). That means the line itself is not part of the solution.
- Shade the region above the line because the inequality includes "greater than or equal to."
- Move to the second inequality: x + 3y ≤ 4
- Convert it to slope-intercept form by isolating y:
y ≤ (-1/3)x + 4/3
- Graph this inequality by drawing a solid line with a slope of -1/3 and a y-intercept of 4/3. Make the line solid because the inequality is less than or equal to (≤). That means the line itself is part of the solution.
- Shade the region below the line because the inequality includes "less than or equal to."
- Finally, graph the last inequality: x ≥ -5
- This is a simple vertical line going through the x-coordinate -5. Since the inequality includes "greater than or equal to," the line itself is part of the solution.
- Shade the region to the right of the line.
Step 2: Determine the overlapping shaded regions.
- The solution to the system of inequalities is the region where all the shaded regions overlap. It represents the values that satisfy all three inequalities simultaneously.
Step 3: Label the solution set, if required.
- If you need to label the solution set, include an appropriate description or note with the graph.
This process will help you graph the solution for any system of linear inequalities.