Foundations of Math 11
PRACTICE Unit Exam: Linear Inequalities/Linear Programming
1. Determine whether the given ordered pair is a solution to the following equation. Show
your work:
1
a) (2, 3); 3𝑥 − 5𝑦 = −9
Yes/No
b) (0, 4); 𝑦 = − 3 𝑥 + 4
Yes/No
c) (1, −1); 3𝑦 = 5 − 2𝑥
Yes/No
d) (−1, 3); 𝑦 = −1
Yes/No
2. Graph the following equations. Assume 𝑥 ∈ ℝ, 𝑦 ∈ ℝ.
a) 𝑦 = 3𝑥 − 2
Slope: ___ y-int: ____
b) 2x + y = −4
Slope: ___ y-int: ____
c) 3𝑥 + 2𝑦 = 10
Slope: ___ y-int: ____
d) 𝑦 = −3
Slope: ___ y-int: ____
3. Determine the following for each linear inequality:
i) the equation of the boundary line;
ii) whether the boundary line is solid or dashed;
iii) if (0, 0) is a solution;
iv) whether the half plane is shaded above or below the boundary line.
a) 3𝑥 + 2𝑦 > 18
b) −𝑦 ≤ 3𝑥 − 9
c) 𝑥 − 6 ≥ 2𝑦
i)
i)
i)
ii)
ii)
ii)
iii)
iii)
iii)
iv)
iv)
iv)
4. Match the graph with the system described below.
1.
1
{(𝑥, 𝑦)|𝑦 ≥ 𝑥 + 5, 𝑥 ∈ ℝ, 𝑦 ∈ ℝ}
3
1
2. {(𝑥, 𝑦)|𝑦 ≥ 3 𝑥 + 5, 𝑥 ∈ 𝑊, 𝑦 ∈ 𝑊}
3.
A)
1
{(𝑥, 𝑦)|𝑦 > 𝑥 + 5, 𝑥 ∈ ℕ, 𝑦 ∈ ℕ}
3
B)
C)
5. Graph and shade these inequalities. Assume 𝑥 ∈ ℝ, 𝑦 ∈ ℝ.
3
a) 𝑦 < − 4 𝑥 + 1
Slope: ___ y-int: ____
b) x ≥ 3
c) 5𝑥 − 7𝑦 < −14
Slope: ___ y-int: ____
d) 𝑦 ≤ 3 𝑥
Slope: ___ y-int: ____
f) 𝑥 + 2𝑦 ≥ 6
Slope: ___ y-int: ____
e) 𝑥 + 𝑦 < 8
Slope: ___ y-int: ____
1
Slope: ___ y-int: ____
6. Graph and shade (or stipple) appropriately:
1
a) {(𝑥, 𝑦)| 𝑦 > − 4 𝑥 + 8, 𝑥 ∈ ℝ, 𝑦 ∈ ℝ }
1
1
b) {(𝑥, 𝑦)| 𝑦 > − 4 𝑥 + 8, 𝑥 ∈ 𝑊, 𝑦 ∈ 𝑊 } c) {(𝑥, 𝑦)| 𝑦 > − 4 𝑥 + 8, 𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 }
7. Graph the systems of linear inequalities; shade in the solution. Assume 𝑥 ∈ ℝ, 𝑦 ∈ ℝ.
a) 𝑦 ≤ 3𝑥 − 4
&
5
𝑦 ≥ −4𝑥 + 5
b) 𝑥 + 𝑦 ≤ 2
&
𝑥 + 𝑦 ≥ −2
c) 𝑦 < 5
d) 4𝑥 + 5𝑦 < 20
5
𝑦 > −4𝑥 + 5
3
𝑦 > 4𝑥 − 3
2𝑥 − 𝑦 ≤ 4
𝑥 ≥ 0, 𝑦 ≥ 0
8. Check all vertices of the feasible region to determine the maximum and minimum values of
objective function = 3𝑥 + 2𝑦 , subject to the following constraints:
𝑥 − 2𝑦 ≤ 4
𝑥+𝑦 ≤1
Max value: C = _______ at point (
,
𝑥≥0
Min value: C = _______ at point (
,
)
)
9. Check all vertices of the feasible region to determine the maximum and minimum values of
objective function 𝐶 = 3𝑥 + 2𝑦 , subject to the following constraints:
4𝑥 + 5𝑦 ≥ 100
3𝑥 + 4𝑦 ≤ 240
Max value: C = _______ at point (
,
𝑥 ≤ 60
𝑦 ≤ 45
Min value: C = _______ at point (
,
𝑥≥0
𝑦≥0
)
)
10. A farmer has 36 acres of land for planting wheat and corn. The cost and time are listed
below. Find the maximum profit.
Preparation cost/acre
Work per acre
Profit per acre
a) Define your variables:
Wheat
$60
3 hours
$180
Corn
$30
4 hours
$100
Maximum
$1800
120 hours
Let x = ___________________ and y = _________________
and 𝑥_____, 𝑎𝑛𝑑 𝑦______
b) Write two inequalities from the information given in the chart:
Prep/Cost Parameter:
Work Parameter:
c) Sketch and shade the four inequalities on the grid provided below.
d) Write an objective function for maximizing profit: P =
e) Test ALL vertices of the feasible region to find the highest profit.
f) Write a solution statement: