# 5.5 – Two-Variable Systems of Inequalities

Key Terms

• Axes – The reference lines used in a graph.
• Axes is the plural of axis.
• The axes of a coordinate graph are perpendicular, which means they intersect at 90° angles.
• Axis – A reference line used in a graph.
• The x-axis is horizontal, and the y-axis is vertical.
• Boundary Line – A solid or dashed line on a graph that provides a border for the solution set.
• Systems of Inequalities – Groups of inequalities that have the same variables and are used together to solve a problem.

Review

• Greater Than:  >
• Numbers of greater value
• Numbers to the RIGHT on the number line
• Less Than: <
• Numbers of lesser value
• Numbers to the LEFT on the number line
• Greater Than or Equal to: ≥
• Numbers of greater value OR equal value
• “At Least”
• Less Than or Equal to: ≤
• Numbers of lesser value OR equal value
• “At Most”

• Graphing Linear Equations
• Slope-Intercept Form: y = mx + b
• m = slope (rise / run)
• b = y-intercept (0, b)
• Example: y = x + 2
• Slope: 1
• y-intercept (0, 2)

Notes

• Graphing Linear Inequalities
• Graph lines in slope-intercept form: y = mx + b
• Make the line dashed if y is > or < the line (mx + b)
• Make the line solid if y ≥ or ≤ the line (mx + b)
• Shade the half-plane that contains the solutions
• If  y is GREATER than the line (mx + b): y > mx + b, shade ABOVE the line and make sure the line is DASHED
• If   y is LESS than the line (mx + b): y < mx + b, shade BELOW the line and make sure the line is DASHED
• If   y is GREATER THAN or EQUAL TO the line (mx + b): y ≥ mx + b, shade ABOVE the line and make sure the line is SOLID
• If y is LESS THAN or EQUAL TO the line (mx + b): y ≤ mx + b, shade BELOW the line and make sure the line is SOLID

• Solution Set for a System of Two Inequalities
• All the points in the area where the two graphed half-planes overlap.
• To solve a system of inequalities graphically, you just need to graph each inequality and see which points are in the overlap of the graphs.
• Example: The two lines are graphed and their solutions are shaded.
• The pink is shaded above a dashed line
• The blue is shaded below a solid line
• The purple is the solution set for these two inequalities.
• It includes ALL points on the solid line, but does NOT include any points on the dashed pink line.

• Now, test a point in the overlap (purple) area to see if it works with BOTH inequalities

• Parallel Lines
• No Solution
• Sometimes the graphs will never overlap
• The lines will be parallel
• The top line will have shading above it and the bottom line will have shading below it

• With Solutions
• Parallel lines that have shading on top or on the bottom of BOTH lines will have overlap

Examples

• Ex 1. Which of the following points are solutions to the system of inequalities shown below?
• y > 5x -2
y < 5x +3
• Possibilities: (-5, 25), (4, 20), (5, 28), (4, 23)
• Any points along a DASHED line are NOT part of the solution
• Points MUST be inside the purple (overlap) section or along a SOLID line

• Ex 2. Which of the following points are solutions to the system of inequalities shown below?
• $x+y \leq 4+3$
• $y \textgreater 3$
• Possibilities: (3, 6), (2, -1), (1, 5), (1, 1), (-1, 8), (2, 8)
• Points along the SOLID line ARE part of the solution
• Points in the purple (overlap) section ARE part of the solution

Answers: (1, 5) and (-1, 8)

• Ex 3. The graph below shows the solution to which system of inequalities?
• Step 1: locate the y-intercepts
• The dashed line crosses y at 0
• The solid line crosses y at positive 3
• Step 2: look at the rise over run (slope) for each line
• The dashed line has a slope of $\frac{1}{1}$, so a slope of 1
• The solid line has a zero slope (no rise, all run: horizontal line)
• Step 3: write the equations for each line
• Dashed: y = 1x + 0, which simplifies to y = x
• Solid: y = 0x + 3, which simplifies to y = 3
• Step 4: Look at the shaded area in relation to each line
• Dashed: the shaded area is below it
• Solid: the shaded area is above it
• Step 5: Change the equal signs to inequality signs
• Dashed: y = x becomes y < x (since all y-solutions are below it)
• Solid: y = 3 becomes $y \geq 3$