**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)

- Slope-Intercept Form: y = mx + b

**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

- Graph lines in slope-intercept form: y = mx + b

- 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

- No Solution

- 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

- y > 5x -2

Answer: only (4, 20)

- Ex 2. Which of the following points are solutions to the system of inequalities shown below?
- 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

- Possibilities: (3, 6), (2, -1), (1, 5), (1, 1), (-1, 8), (2, 8)

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 , 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

- Step 1: locate the y-intercepts