**Key Terms**

- xy-plane – A plane formed by a horizontal number line (the x-axis) and a vertical number line (the y-axis) that intersect at the zero point of each.
- The xy-plane is also called the Cartesian coordinate system or coordinate plane.

- Half-Plane – The graph of a linear inequality with two variables.
- If the inequality sign is < or >, the line is dashed to show that it is not included in the solution.
- If the inequality sign is or , the line is solid to show that it is included in the solution.

**Review**

- < is less than
- Ex: 2 < 6, two is less than six

- > is greater than
- Ex: 5 > 3, five is greater than three

- is less than or equal to (is at most…)
- Ex: 4 4, four is less than or equal to four
- Ex: 5 6, five is less than or equal to six
- m 5, Megan is at most 5 feet tall (she could be 5 feet tall, but she could be shorter)

- is greater than or equal to (is at least…)
- Ex: 7 7, seven is greater than or equal to seven
- Ex: 8 7, eight is greater than or equal to seven
- z 9, Zander is at least 9 years old (he may be 9 years old, but he could be older)

- Remember: the sign is like a mouth. The mouth eats more!

- Plotting x and y
- y is vertical (up and down) on an xy-plane
- x is horizontal (left and right) on an xy-plane

- Slope
- Positive slopes rise from left to right
- m = 4, m = 1/2, etc.

- Negative slopes fall from left to right
- m = -5, m= -1/8, etc.

- Perpendicular slopes are negative reciprocals of each other
- Slope

- Steepness
- A steeper slope has a greater number. It doesn’t matter if it’s negative or positive.
- m = -4 is steeper than m = 2 because 4 is bigger than 2.

- Positive slopes rise from left to right

**Notes**

- When working with linear inequalities, you will have to shade above or below the line on the xy-plane.
- This shading is called the half-plane.
- The line will be either dashed or solid depending on the inequality.
- Dashed: < or >
- Solid: or

- You will have to test out some points (using substitution) to see whether or not you shade above or below the line.

- There are actually two ways to determine how to shade the half-plane.
- The first way: Replace, then Check, then Shade
- Replace the inequality symbol with an equals sign and graph the equation as a solid or dashed line.
- Check a point on either side of the line and note which one satisfies the inequality.
- Shade the half-plane that contains a point that satisfies the inequality.

- The first way: Replace, then Check, then Shade

- Example: y > 2x + 3
- Step 1: Draw the line on a graph and REPLACE the > with an equal sign: y = 2x + 3 to help you draw the line.
- Draw a dashed line if the inequality symbol is > or <
- In this case, the original inequality is >, so make it a dashed line (below)

- Step 2: Choose a point on either side of that line to CHECK (test) the inequality
- I chose (2.0) as it’s easy to test a point with a zero in it.
- y > 2x + 3 would be 0 > 2(2) + 3
- Simplify: 0 > 7 is false, so you do not shade near the chosen point. Instead, SHADE on the OTHER side of the line.

- Step 1: Draw the line on a graph and REPLACE the > with an equal sign: y = 2x + 3 to help you draw the line.

- The second way is to look at the inequality after y. Is it greater or less than the line?
- If y > (slope & y-intercept), shade above the dashed line
- y > x
- If y (slope & y-intercept), shade above the solid line
- y x
- If y < (slope & y-intercept), shade below the dashed line
- y < x
- If y (slope & y-intercept), shade below the solid line
- y x

**Examples**

- Ex 1: Change this to slope-intercept form:
*y*– 7 < 4(*x*– 3), draw the line, then shade the half-plane.- Step 1: distribute the 4
*y*– 7 < 4*x*– 12

- Step 2: add 7 to both sides
*y*< 4*x*– 12 + 7

- Step 3: simplify
*y*< 4*x*– 5

- Step 4: Note the slope and y-intercept and plot them on a graph
- Slope: 4 (so rise 4, run 1)
- y-intercept: (0, -5)

- Step 1: distribute the 4

- Step 5: Shade the half-plane
- Since y < the line, shade below the line

- Ex 2. Which of the following points satisfy the inequality:
- Choices:
- A. (0, 0)
- Test: becomes No

- B. (2, -2)
- Test: becomes Yes

- C. (3, -6)
- Test: becomes Yes

- D. (3, -8)
- Test: becomes Yes

- E. (4, -5)
- Test: becomes Yes

- F. (4, -8)
- Test: becomes Yes

- A. (0, 0)

- Ex 3. A group of friends are hiking on trail Y when they reach an intersection with trail X.
- Trail X is perpendicular to trail Y.
- If trail Y has a slope of , which of the following statements are true?
- We know that perpendicular slopes are negative reciprocals
- Fact, trail Y:
- So, trail X would be : , or just 2.
- Trail X is 2, which is a positive slope (it runs uphill).
- Trail X is steeper than trail Y since 2 is a bigger number than 1/2
- Steepness doesn’t matter if it’s negative or positive

- We know that perpendicular slopes are negative reciprocals