Section 2.3. Linear Inequalities

Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities This section covers inequalities. An important skill in this section will be to represent inequalities using interval notation and using number line graphs. The following chart will help. Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities Here are some specific examples of how to represent inequalities using interval notation and using a number line: There are two ways to represent an inequality on a number line. • Open and closed circles • Round parenthesis and brackets This reversal property of inequality will allow us to rewrite inequalities in alternate forms. I use this often when I am asked to sketch a graph on a number line. It is easier to me when the variable is on the left side of the inequality. Original problem Rewrite using the reversal property 12 > 7 7 < 12 9 > 6𝑥 + 3 6𝑥 + 3 < 9 5 ≤ 3𝑥 − 4 3𝑥 − 4 ≥ 5 𝑥≥3 3≤𝑥 Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities 1) Complete the table Inequality Graph Interval Notation 𝑥 > −3 𝑥≤5 −2 ≤ 𝑥 < 2 (5, ∞) (−∞, −3) [−2,5) Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities 2) Complete the table Inequality Graph Interval Notation 𝑥≥6 𝑥<2 −4 < 𝑥 < 6 (−∞, 2) (5, ∞) (−2,5] Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities 3) Complete the table Inequality 𝑥 > −∞ Graph Interval Notation This is tricky. 𝑥≥5 −2 < 𝑥 ≤ 2 (−4, ∞) (−∞, −6] [1,5] Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities 4) Complete the table Inequality 𝑥<∞ Graph Interval Notation This is tricky. 𝑥≥7 −1 ≤ 𝑥 < 4 (−∞, 0] (−2, ∞) [1,7] Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities The addition and subtraction property of inequality allow us to isolate a variable by adding and or subtracting in a problem with an inequality without changing the direction of the sign. Examples of using the addition property of inequality. Examples of using the subtraction property of inequality. Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities The multiplication and division property of inequality allow us to isolate a variable by multiplying or dividing both sides of an inequality by the same number. • Switch the direction of the sign if multiply or divide by a negative number • Keep the direction of the sign if you multiply or divide by a positive number When you multiply or divide by a negative number you change the direction of the sign When you multiply or divide by a positive number, we keep the direction of the sign. 5 𝑆𝑜𝑙𝑣𝑒 − 4 𝑥 = −10 (flip the sign since you need to multiply by a negative to isolate the x.) Number line answer: As in interval (8, ∞) Number line answer: Interval answer: [3, ∞) Notice the sign does not flip since a positive number was divided on both sides. Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities For example: Solve, express each answer using an inequality, a graph, and interval notation. −5𝑥 + 10 < −15 Step 1: nothing needed, each side is simplified. Step 2: nothing needed, there are not variables on each side of the inequality Step 3: −5𝑥 + 10 < −15 −10 − 10 −5𝑥 < −25 Step 4: −5𝑥 −5 > −25 −5 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑖𝑔𝑛 𝑔𝑒𝑡𝑠 𝑓𝑙𝑖𝑝𝑝𝑒𝑑 𝑠𝑖𝑛𝑐𝑒 𝐼 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟. Answer: 𝑥 > 5 (written as an inequality) Answer in interval notation: (5, ∞) Answer written on a number line. To check, pick one number less than five and one greater than 5. Plug each number into the original inequality. The number that is part of the solution should produce a true statement. The number that is not part of the solution should produce a false statement. 𝐼 𝑝𝑖𝑐𝑘 𝑥 = 4 𝑎𝑛𝑑 𝑥 = 6 I should get a false statement when I substitute 𝑥 = 4, since 4 is not part of the solution, −5(4) + 10 < −15 −20 + 10 < −15 −10 < −15 This is not true, and this is what should happen. I should get a true statement when I substitute x = 6, since 6 is part of the solution. −5(6) + 10 < −15 −30 + 10 < −15 −20 < −15 This is true, and this is what should happen. Both tests produced the required results. This is the best I can do to check my work. ✓ Section 2.3: Linear inequalities Chapter 2: Functions, Linear equations, and inequalities General strategy for solving linear equalities. 1) Simplify each side of the equation as much as possible. • Use the Distributive Property to remove any parentheses. • Combine like terms when needed. 2) Collect all the variable terms on one side of the equation. • Add or subtract as needed 3) Collect all the constant terms on one side of the equation. • Add or subtract as needed 4) Make the coefficient of the variable term equal to 1. • Divide if the coefficient of the variable is an integer. (Remember to switch direction of the sign if you divide by a negative number) • Multiply by the reciprocal if the coefficient of the variable is a fraction. (Remember to switch direction of the sign if you multiply by a negative number) 5) Check the solution. • Substitute two check numbers into the original problem, one that should work and the other that should not work. #1-24: Solve each inequality. Express each answer using an inequality, a graph, and interval notation. 5) 𝑥 + 5 < 10 6) 𝑦 + 6 < 14