Linear Inequalities Exercise 3
Question:
Solve the following system of inequalities graphically 2x+y≥6,3x+4y≤12
Answer:
Step 1: Plot the two lines on the graph.
Step 2: Shade the region where the two lines intersect.
Step 3: The solution to the system of inequalities is the shaded region.
Question:
Solve the given inequalities graphically: x+y≤ 6,x+y ≥ 4
Answer:
Step 1: Draw the xaxis and yaxis on the graph paper.
Step 2: Plot the given inequalities on the graph paper.
Step 3: Shade the region satisfying the inequality, i.e., the region where both the inequalities are true.
Step 4: The solution of the given system of inequality is the shaded region.
Question:
Solve the following system of inequalities graphically 3x+4y≤60,x+3y≤30,x≥0,y≥0
Answer:

Graph the lines 3x+4y=60, x+3y=30, x=0, and y=0 on a coordinate plane.

Shade the area of the coordinate plane that satisfies all of the inequalities.

The solution set is the area of the coordinate plane that is shaded.
Question:
Solve the system of inequalities graphically 3x+2y≤150,x+4y≤80,x≤15,y≥0,x≥0
Answer:
Step 1: Plot the lines 3x+2y=150, x+4y=80, x=15, y=0, and x=0.
Step 2: Shade the region that satisfies all the conditions.
Step 3: The solution is the shaded region.
Question:
Solve the given inequalities graphically: x+2y≤10,x+y≥1,x−y≤0,x≥0,y≥0
Answer:
Step 1: Draw the xaxis and yaxis on the graph.
Step 2: Plot the line x+2y=10 and find the region where x+2y≤10.
Step 3: Plot the line x+y=1 and find the region where x+y≥1.
Step 4: Plot the line x−y=0 and find the region where x−y≤0.
Step 5: Plot the line x=0 and find the region where x≥0.
Step 6: Plot the line y=0 and find the region where y≥0.
Step 7: Shade the region where all the inequalities are satisfied. This is the solution of the given inequalities.
Question:
Solve the following inequations graphically: 2x−y>1,x−2y<−1
Answer:
Step 1: Plot the equations on the graph.
Step 2: Draw a line for each equation.
Step 3: Shade the region that satisfies both equations.
Step 4: The solution set is the shaded region.
Question:
Solve the following system of inequalities graphically x≥3,y≥2
Answer:
Step 1: Draw the coordinate plane.
Step 2: Draw the line x=3 and y=2 on the coordinate plane.
Step 3: Shade the area of the coordinate plane that satisfies both inequalities. This area is the solution.
Question:
Solve: x+y≥4, 2x−y>0
Answer:
Step 1: Subtract y from both sides of the first equation: x ≥ 4  y
Step 2: Add y to both sides of the second equation: 2x > y
Step 3: Subtract x from both sides of the second equation: y > 2x  x
Step 4: Simplify the second equation: y > x
Step 5: Substitute 4  y for x in the second equation: y > 4  y
Step 6: Add y to both sides of the second equation: 2y > 4
Step 7: Divide both sides of the second equation by 2: y > 2
Question:
Solve the following system of inequalities graphically : 4x+3y≤60,y≥2x,x≥3,x,x≥0
Answer:

Graph the line 4x+3y=60

Graph the line y=2x

Graph the line x=3

Shade the region where all inequalities are satisfied

The solution set is the region where all inequalities are satisfied and x≥0.
Question:
Solve the system of inequalities graphically: x−2y≤3,3x+4y≥12,x≥0,y≥1
Answer:
Step 1: Graph the lines x−2y=3, 3x+4y=12, x=0, and y=1.
Step 2: Determine the points of intersection of the lines.
Step 3: Shade the region that satisfies all of the inequalities.
Step 4: The solution is the shaded region.
Question:
Solve the following system of inequalities graphically 2x+y≥8,x+2y≥10
Answer:
Step 1: Plot the lines for the two equations on the same graph:
2x + y = 8 x + 2y = 10
Step 2: Find the points of intersection of the two lines.
The points of intersection are (4, 4) and (6, 2).
Step 3: Shade the region that satisfies both equations.
The shaded region represents the solution to the system of inequalities.
Question:
Solve the given inequalities graphically: 3x+2y ≤ 12,x ≥ 1,y ≥2
Answer:
Step 1: Draw the coordinate plane.
Step 2: Plot the points (1,2) and (0,6).
Step 3: Draw the line through the two points.
Step 4: Shade the region above the line.
Step 5: The solution set is the shaded region.
Question:
Solve the given inequalities graphically: x+y≤9,y>x,x≥0
Answer:

Plot the line x + y = 9.

Shade the area below the line and to the right of the xaxis.

Shade the area below the line and to the left of the yaxis.

The solution is the area not shaded, which is the area above the line and to the right of both axes.
Question:
Solve the system of inequalities graphically: 5x+4y≤ 20,x≥1,y ≥2
Answer:
Step 1: Graph the inequalities 5x+4y≤ 20 and x≥1 on the same coordinate plane.
Step 2: Shade the region that satisfies both inequalities.
Step 3: Graph the inequality y ≥2 and find the points of intersection with the other two inequalities.
Step 4: Shade the region that satisfies all three inequalities.
Question:
Solve the following system of inequalities graphically 2x+y≥4,x+y≤3,2x−3y≤6
Answer:
Step 1: Graph the lines 2x + y = 4, x + y = 3, and 2x  3y = 6 on the coordinate plane.
Step 2: Determine the points of intersection of the lines.
Step 3: Shade the region that satisfies all three inequalities. This is the solution set.