Graphing Linear Inequalities

Show your work for each problem.

1.
What type of line do you use for ?
1. a)A curved line
2. b)No line needed
3. c)Dashed line (points on the line are NOT solutions)
4. d)Solid line (points on the line ARE solutions)
Answer: Dashed line (points on the line are NOT solutions)
Since the symbol is (greater than, not greater than or equal to), points ON the boundary line are NOT solutions. Use a dashed line to show this.
2.
What type of line do you use for ?
1. a)No line, just shading
2. b)A wavy line
3. c)Solid line (points on the line ARE solutions)
4. d)Dashed line (points on the line are NOT solutions)
Answer: Solid line (points on the line ARE solutions)
Since the symbol is (less than or equal to), points ON the boundary line ARE solutions. Use a solid line to show they're included.
3.
What is the slope of the boundary line for ?
Answer: 3
The inequality is already in slope-intercept form. The coefficient of is 3, so the slope is 3.
4.
What is the y-intercept of the boundary line for ?
Answer: 7
The inequality has a boundary line . The y-intercept is the constant 7.
5.
For the inequality , is the point a solution?
1. a)Yes, because the origin is always a solution
2. b)Cannot determine without graphing
3. c)No, because is false
4. d)Yes, because is true
Answer: No, because is false
Substituting : . This is FALSE, so is NOT a solution. Shade the opposite side of the line from the origin.
6.
Which region should be shaded for after testing ?
1. a)Both sides of the line
2. b)The side NOT containing because the test failed
3. c)Neither side, only the line
4. d)The side containing because the test passed
Answer: The side NOT containing because the test failed
Testing : . FALSE! Since the origin doesn't satisfy the inequality, shade the OPPOSITE side (below the line, away from the origin).
7.
Graph the inequality . Determine the boundary line, line type, and which side to shade.
Answer: dashed, above
1. What is the equation of the boundary line? y = -x + 4
2. What is the slope of this line? -1
3. Should the line be solid or dashed? dashed
4. Test : Is ? Is ? false
5. Since failed, shade above or below the line? above
8.
Convert to slope-intercept form. What is the y-intercept?
Answer: 6
. The y-intercept is 6.
9.
Graph . Convert to slope-intercept form first, then determine line type and shading.
Answer: solid, below
1. Subtract from both sides: . Now divide by . What happens to the inequality? flips
2. After dividing by , what is the inequality? y <= 2x - 4
3. Should the line be solid or dashed? solid
4. Test : Is ? Is ? false
5. Since failed, shade above or below? below
10.
The boundary line passes through the origin. Which test point should you use instead of ?
1. a)A point on the line itself
2. b) still works
3. c)No test point is needed
4. d) or - any point NOT on the line
Answer: or - any point NOT on the line
When the boundary passes through , you can't use the origin as a test point. Choose any other simple point like or that's clearly not on the line.
11.
Graph completely. Note: the boundary passes through the origin.
Answer: solid, above
1. What is the slope of the boundary line? 3
2. Does the line pass through the origin? yes
3. Should the line be solid or dashed? solid
4. Can you use as a test point? no
5. Test : Is ? Is ? false
6. Since is below the line and failed, shade above or below? above
12.
For the inequality , convert to slope-intercept form. What is the slope?
Answer: 2
. The slope is 2. (Note: dividing by positive 2 doesn't flip the inequality.)
13.
A company has at most 500 dollars for supplies. Paper costs 5 dollars per ream and ink costs 25 dollars per cartridge. Which inequality represents this?
1. a)
2. b)
3. c)
4. d)
Answer:
'At most 500 dollars' means the total can be 500 or less, so we use . Cost = .
14.
You need to buy at least 12 notebooks. Large notebooks cost 8 dollars and small ones cost 4 dollars. Your budget is 60 dollars. Write an inequality for the budget constraint and test if 5 large and 4 small notebooks is affordable.
Answer: yes
1. Let = large notebooks, = small notebooks. What is the budget inequality? 8L + 4S <= 60
2. Calculate the cost: 56
3. Is ? yes
4. How many total notebooks is that? 9
5. Does this satisfy the 'at least 12' requirement? no
15.
What does the shaded region represent in a graph of a linear inequality?
1. a)Points that are NOT solutions
2. b)Only the points on the boundary line
3. c)The origin only
4. d)All the points that are solutions to the inequality
Answer: All the points that are solutions to the inequality
The shaded region (called a half-plane) contains ALL the ordered pairs that make the inequality TRUE. Every point in the shaded area is a solution.