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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: