LESSON 1-5 NOTES (Part A): SOLVING INEQUALITIES Words like "at most" and "at least" suggest a relationship in which two quantities may not be equal. These relationships can be represented by a mathematical inequality. Inequality symbols include less than (<), less than or equal to ( < ), great than ( >), greater than or equal to ( > ), or not equal to ( ). REVIEW: Write each statement as an inequality and then graph the inequality. Statement Inequality Graph x is greater than 4 x is greater than or equal to x is less than 4 x is less than or equal to 4 Open dot Closed dot Value is not a solution to the inequality. Value is one of the solutions to the inequality. EXAMPLE 1: What inequality represents the sentence "5 fewer than a number is at least 12."? PRACTICE: Write the inequality that represents the sentence. 1) Five less than a number is at least 28. 2) The product of a number and four is at most 10. 3) Six more than a quotient of a number and three is greater than 14. As with an equation, the solutions of an inequality are numbers that make it true. The procedure for solving a linear inequality is much like the one for solving linear equations. To isolate the variable on one side of the inequality, perform the same algebraic operation on each side of the inequality symbol. Below are some properties of inequalities to keep in mind when solving inequalities. The Addition and Subtraction Properties of Inequality state that adding or subtracting the same number from both sides of the inequality does not change the inequality. If a < b, then a + c < b + c. If a < b, then a c < b c. The Multiplication and Division Properties of Inequality state that multiplying or dividing both sides of the inequality by the same positive number does not change the inequality. If a < b and c > 0, then ac < bc. If a < b and c > 0, then a c < b c. * The procedure for solving an inequality is similar to the procedure for solving an equation but with one important exception. The Multiplication and Division Properties of Inequality also state that, when you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol. If a < b and c < 0, then ac > bc. If a < b and c < 0, then a b c c.
* Remember, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. Example: 5 > 3 multiply both sides by -2 5. (-2) < 3. (-2) -10 < - 6 EXAMPLE 2: What is the solution of 3(x + 2) 5 21 x? Graph the solution. Justify each line in the solution by naming one of the properties of inequalities. 3x + 6 5 21 x Distributive Property 3x + 1 21 x Simplify. 4x + 1 21 Addition Property of Inequality 4x 20 Subtraction Property of Inequality x 5 Division Property of Inequality To graph the solution, locate the boundary point. Plot a point at x = 5. Because the inequality is "less than or equal to, the boundary point is part of the solution set. Therefore, use a closed dot to graph the boundary point. Shade the number line to the left of the boundary point because the inequality is less than. Graph the solution on a number line. EXAMPLE 3: What is the solution of 2x 3(x 1) < x + 5? Graph the solution. Justify each line in the solution by naming one of the properties of inequalities. 2x 3(x 1) < x + 5 2x 3x + 3 < x + 5 Distributive Property x + 3 < x + 5 Simplify. 2x < 2 Subtraction Property of Inequality x > 1 Division Property of Inequality The direction of the inequality changed in the last step because we divided both sides of the inequality by a negative number. Graph the solution on a number line. PRACTICE: Solve each inequality. Graph the solution. 4) -3x + 5 17 5) 12 + 5x < -23 6) 2x + 4(x 2) > 4 7) 4 (2x 4) 5 (4x + 3)
LESSON 1-5 NOTES: SOLVING INEQUALITIES EXAMPLE: Using inequalities to solve a problem (Problem 3, Page 35 of textbook) A movie rental company offers two subscription plans. You can pay $36 a month and rent as many movies as desired, or you can pay $15 a month and $1.50 to rent each movie. How many movies must you rent in a month for the first plan to cost less than the second plan? Define the variable. Write an expression for the cost of each plan. Write an inequality for the problem situation. Solve the inequality. Write the answer in a complete sentence. PRACTICE 1: (Got It? #3, Page 35 of textbook) A digital music service offers two subscription plans. The first has a $9 membership fee and charges $1 per download. The second has a $25 membership fee and charges $0.50 per download. How many songs must you download for the second plan to cost less than the first plan? Define the variable. Write an expression for the cost of each plan. Write an inequality for the problem situation. Solve the inequality. Write the answer in a complete sentence. PRACTICE 2: The width of a rectangle is 4 cm less than the length. The perimeter is at most 48 cm. What are the restrictions on the dimensions of the rectangle? Solve the problem using an inequality.
LESSON 1-5 NOTES (Part B): SOLVING INEQUALITIES INEQUALITIES WITH NO SOLUTION -OR- ALL REAL NUMBERS AS SOLUTIONS EXAMPLES: Is the inequality always, sometimes or never true? (Problem 4, Pg. 36 of text) A) -2(3x + 1) > -6x + 7 B) 5(2x - 3) - 7x 3x + 8 PRACTICE: Is the inequality always, sometimes, or never true? 3) 5(x 2) 2x + 1 4) 2x + 8 2(x + 1) 5) 6x + 1 3(2x 4) 6) 2(3x + 3) 2(3x + 1) COMPOUND INEQUALITY: Two inequalities can be combined with the word and or the word or to form a compound inequality. Compound and inequalities can be combined into a simpler form. For example, the and inequality 5 < x + 1 and x + 1 < 13 can be written as 5 < x + 1 < 13. It is read as " x + 1 is greater than 5 and less than 13." Graphs of compound and inequalities are "segments." The graphs of compound or inequalities are "rays" with different endpoints that go in opposite directions. EXAMPLES: SOLVING AN "AND" INEQUALITY - To solve a compound inequality containing and, find all the values that make both inequalities true. Find and graph the solution to each compound and inequality. A) 7 < 2x + 1 and 3x 16 B) -3 < 2x + 5 7
PRACTICE: Find and graph the solution to each compound and inequality. 7) 3x - 1 5 and 2x < 12 8) -5 x - 3 < 2 EXAMPLE: SOLVING AN "OR" INEQUALITY - To solve a compound inequality containing or, find all the values of the variable that make at least one of the inequalities true. Find and graph the solution to: 7 + k 6 or 8 + k < 3 PRACTICE: Find and graph the solution to each compound or inequality. 9) 7w + 3 > 11 or 4w - 1 < -13 10) 16 < 5x + 1 or 3x + 9 < 6 PRACTICE: Write a compound inequality to represent each sentence. 10) The average of Shondra s test scores in Physics is between 88 and 93. 11) The Morgans are buying a new house. They want to buy either a house more the 75 years old or a house less than 10 years old.