# 4.6 – Linear Inequalities

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 $\leq$ or $\geq$, 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
• $\leq$ is less than or equal to (is at most…)
• Ex: 4 $\leq$ 4, four is less than or equal to four
• Ex: 5 $\leq$ 6, five is less than or equal to six
• $\leq$ 5, Megan is at most 5 feet tall (she could be 5 feet tall, but she could be shorter)
• $\geq$ is greater than or equal to (is at least…)
• Ex: 7 $\geq$ 7, seven is greater than or equal to seven
• Ex: 8 $\geq$ 7, eight is greater than or equal to seven
• $\geq$ 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 $-\frac{1}{4}\perp \frac{4}{1}$
• 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.

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: $\leq$ or $\geq$
• 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
1. Replace the inequality symbol with an equals sign and graph the equation as a solid or dashed line.
2. Check a point on either side of the line and note which one satisfies the inequality.
3. Shade the half-plane that contains a point that satisfies the inequality.

• 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.

• 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 $\geq$ (slope & y-intercept), shade above the solid line
• y $\geq$ x
• If y < (slope & y-intercept), shade below the dashed line
• y < x
• If y $\leq$ (slope & y-intercept), shade below the solid line
• y $\leq$ 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 < 4x – 12
• Step 2: add 7 to both sides
• y < 4x – 12 + 7
• Step 3: simplify
• y < 4x – 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 5: Shade the half-plane
• Since y < the line, shade below the line

• Ex 2. Which of the following points satisfy the inequality: $y\leq 3x-4$
• Choices:
• A. (0, 0)
• Test: $0\leq 3(0)-4$ becomes $0\leq -4$  No
• B. (2, -2)
• Test: $-2\leq 3(2)-4$ becomes $-2\leq 2$  Yes
• C. (3, -6)
• Test: $-6\leq 3(3)-4$ becomes $-6\leq 5$  Yes
• D. (3, -8)
• Test: $-8\leq 3(3)-4$ becomes $-8\leq 5$  Yes
• E. (4, -5)
• Test: $-5\leq 3(4)-4$ becomes $-5\leq 8$  Yes
• F. (4, -8)
• Test: $-8\leq 3(4)-4$ becomes $-8\leq 8$  Yes

• 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 $-\frac{1}{2}$, which of the following statements are true?
• We know that perpendicular slopes are negative reciprocals
• Fact, trail Y: $-\frac{1}{2}$
• So, trail X would be : $\frac{2}{1}$, 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