Download In chapter 3 we gained experience in writing linear equations with

Activity 4: Linear Inequalities
Graphing Solutions
Part I: Review – Graphing linear functions
Graph each of the following equations using either the y-intercept and slope given y = mx + b, or the x-intercept
and y-intercept given Ax + By = C.
1.
y = 2x – 4
2. 3x – 6y = 24
x-intercept:
y – intercept:
Part II: Graphing Linear Inequalities
Important decisions in business often involve many variables and relations among those variables. The key to
making good decisions is finding a way to organize and compare options. Earlier in Chapter 7, you learned
ways to solve systems of linear equations (two or more equations). That is, you found where the two equations
had the same solution (a point).
In this activity we will be working with linear inequalities and finding where these solutions occur
graphically(multiple points). Let us look at a scenario where a summer festival sells tickets in advance and at
the gate. The tickets are $8 if bought in advance and $12 if bought at the gate.
1.
Write a linear equation that represents the income made by the ticket sales. Define your variables.
A=
G=
Equation: Income =
If expenses for operating the two-day festival total $2,400, then solutions of 8A + 12G = 2400 give (A, G)
combinations for which festival income will equal operating expenses.
2. Rewrite the previous equation as an inequality where ….
i. the festival loses money (i.e. expense of 2400 is greater that income)
ii. the festival makes money.
Activity 4: Linear Inequalities
Graphing Solutions
3. The diagram below gives a graph of the line
representing solutions for the equation 8A + 12G = 2400
(in the first quadrant only). Use the graph as a starting
point in solving the inequality 8A + 12G >2400.
a. For each point on the diagram, give the (A, G)
coordinates and the festival income from that
combination of tickets sold in advance and tickets
sold at the gate. (share the work within your group).
Point
A
B
C
Coordinates
Income
Point
D
E
F
Coordinates
Income
Point
G
H
I
Coordinates
Income
b. Based on your work in part a, indicate which of the points (A, G) match up with the two inequalities:
i.
8A + 12G > 2400
iii. 8A + 12G < 2400
c. Shade the area of the graph above where the festival will make money.
4. A local bank is giving away souvenir Frisbees and sun visors at the music festival. The Frisbees will cost
the bank $4 each, and the visors will cost the bank $2.50 each. The promotional cost for the bank depends
on the number of Frisbees (F) and the number of Visors (V) given away at the festival. The bank has
budgeted $1000 for the purchase of these items.
a. Write an equation that would represent the combination of Frisbees and Visors that would cost the bank
exactly $1000.
b. Graph the equation on the grid below (1st quadrant only).
c. Write an inequality that represents the question, “How
many souvenir Frisbees and visors can the bank give away
for a total cost of no more than $1000?”
Activity 4: Linear Inequalities
Graphing Solutions
d. Find two points (F, V) where the cost to the bank would be greater than $1000.
e. Find two points (F, V) where the cost to the bank would be less than $1000.
f. Shade the region of the graph where the bank would have a cost of less than $1000 for their Frisbees and
Visors handed out.
The previous problems had only meaningful solutions of inequalities in the first quadrant. Negative values of
A, G, F, and V make no sense in those situations. However, the ideas and graphing techniques you developed
can be applied to situations without the constraint of only positive values.
5. For example, the following graph show solutions with both
positive and negative values for x + 2y < -2 and x + 2y = -2.
a. Which region corresponds to solutions of x + 2y < -2? How
did you decide?
b. Which region corresponds to solutions of x + 2y = - 2? How
did you know?
Just as with inequalities in one variable, the solution region of an
inequality in two variables is generally bounded by the solution of a corresponding equation. Sometimes the
boundary is included in the solution of the inequality, indicated on the graph by a solid boundary. A dashed
boundary on a graph indicates that the points on the boundary are excluded from the solution.
6. Given the above information, graph the following inequalities using the following steps.
i. Find the key points needed to graph the equation of the line represented by the inequality.
ii. Decide whether your line should be solid or dashed (as stated above).
iii. Determine whether the points above or below the line satisfy the inequality and shade that region.
a.
c.
3x – 2y < 12
b. 2x + y > 4
Activity 4: Linear Inequalities
c.
7.
8x – 5y > 20
Graphing Solutions
d. 4x + 3y < 15
Suppose the organizers of the music festival can sell more no more than 1000 admission tickets.
a. Write an inequality that represents solutions where the
number of tickets sold will be no more than 1000.
b. Graph this on the coordinate grid provided and shade
the appropriate region that represents where the
inequality is true.
8.
Suppose the bank wants to give away at least 300 promotional items at the festival.
a. Write an inequality whose solutions are the (F, V)
combinations for which the total number of
promotional items will be 300 or more.
b. Draw a graph that shows the region containing all the
points that satisfy the inequality from part a.
Activity 4: Linear Inequalities
Graphing Solutions