WARM UP Solve: 1. 3x – 5 = (3x -5) = x – 3 + 4x = (2x – 4) = 6

INEQUALITIES

OBJECTIVES  Determine if a number is a solution of an inequality and graph the solution set.  Solve inequalities using the addition property  Solve inequalities using the multiplication property  Solve inequalities using both the addition and multiplication properties.

VOCABULARY  Is less than  Is greater than  Inequalities  Solution set  Set-builder notation  Addition property  Addition property of Inequality  Multiplication property  Multiplication property of Inequality

SOLVING INEQUALITIES  If a number occurs to the left of another on the number line, the first number is less than the second, and the second is greater than the first.  The order of the real numbers is often pictured on a number line.  We use the symbol to mean “is greater than.”  The symbol ≤ means “is less than or equal to,” and the symbol ≥ mean is greater than or equal to.”

SOLVING INEQUALITIES  A solution of an inequality is any number that makes it true.  Mathematical sentences containing, ≤, ≥, are called inequalities.  The set of all solutions is called the solution set.  Example: Determine whether the given number is a solution of the inequality x + 3 < 6; 5 We substitute and get 5 + 3< 6 or 8 < 6, a false sentence. Thus 5 is not a solution.

MORE EXAMPLES 2x – 3 > -3, 1 We substitute x and get 2(1) – 3 > -3 or -1 > -3, a true sentence. Thus, 1 is a solution. 4x – 1 ≤ 3x + 2, 3 We substitute and get 4(3) – 1 ≤ 3(3) or 11 ≤ 11, a true sentence. Thus 3 is a solution.

TRY THIS…. 1.3 – x < 2; 4 2.3y + 2 > -1, x + 2 ≤ 4x – 3, 5 Determine whether the given number is a solution of the inequality: We substitute 4 and get 3 – 4 < 2 or -1 < 2, a true statement, therefore 4 is a solution. We substitute -2 and get 3(-2) + 2 > -1 or -4 > -1 a false statement, therefore 4 is not a solution. We substitute 5 and get 3(5) + 2 ≤ 4(5) -3, or 17 ≤ 17 a true statement, therefore 5 is a solution.

GRAPH INEQUALITIES  A graph of an inequality is a drawing that shows all of its solutions on a number line. The graph is a picture of the solution set. Example: Graph x < 2 on a number line. The solution set graphed can be written {x|x < 2]}. This is called set-builder notation. The notation is read “the set of all x such that x is less than 2.” Set notation of this type is written using braces. The symbol | is read “such that.”

EXAMPLE Graph x ≥ -3 on a number line This time the solution set consists of all the numbers greater than -3, including -3. We shade all the numbers greater than -3 and use a solid circle at -3 to indicate that it is also a solution. We draw a picture of the solutions {x|x ≥ -3}

TRY THIS…. Graph on a number line: 1.x < x ≥ 1 3. x ≤ 5

THE ADDITION PROPERTY  Addition Property of Inequality If a < b is true, then a + c < b + c is true for any real number c Similar statements hold for >, ≤, ≥.  If any number is added to both sides of a true inequality, another true inequality is obtained.  To solve an inequality using the addition property, we transform the inequality into a simpler one by adding the same number to both sides, as we do in solving equations.

EXAMPLE Using the addition property adding -4 Solve. Then graph x + 4 > 7 x (-4) > 7 + (-4) x > 3 The solution set is {x|x > 3}. Every number greater than 3 was a solution. Because there are many solution to an inequality, we cannot check all the solutions by substituting into the original inequality as we do for equations. We can check our calculations by substituting for the variable in the original inequality at least one number from the proposed solution set.

TRY THIS…. 1. x + 6 > ≥ x x – 1 ≤ 2x - 3

THE MULTIPLICATION PROPERTY  Consider the true inequality 4 < 9.  If we multiply both numbers by 2, we get the true inequality 8 < 18  If we multiply both numbers by -3, we get the false inequality -12 < -27.  If we reverse the inequality symbol, we get the true inequality -12 > -27.  If we multiply both sides of a true inequality by a positive number, we get another true inequality.  If we multiply both sides of a true inequality by a negative number and reverse the inequality symbol, we get another true inequality.

MULTIPLICATION PROPERTY OF INEQUALITY If a < b is true, then ac < bc is true for any positive real number c, and ac > bc is true for any negative real number c. Similar statements hold for >, ≤, ≥. When we solve an inequality using the multiplication property, we can multiply by any number except zero.

EXAMPLES Multiplying by Any number less than is a solution. The solution set is Solve: 3y < Solve: -4y < Multiplying by and reversing the inequality sign. Any number greater than is a solution. The solution set is

USING THE PROPERTIES TOGETHER  We use addition and multiplication properties together in solving inequalities in much the same way as for equations. Example: Solve 16 – 7 y ≥ 10y – 4 Adding -16 Adding -10y Multiplying by and reversing the inequality sign. The solution set is

WARM UP a. 6 – 5y ≥ 7 b. 3x + 5x < 4 c. 17 – 5y ≤ 8y – 5 Solve: -5y > 1 y < -1/5 8y < 4 y > 1/2 -13y < -22 y > 22/13

USING INEQUALITIES  The problem solving guidelines can help when solving solving problems that translate to inequalities rather than to equations.  PROBLEM SOLVING GUIDELINES UNDERSTAND the problem Develop and carry out a PLAN Find the ANSWER and CHECK.

EXAMPLE In a history course there will be three tests. You must get a total score of 270 for an A. You get 91 and 86 on the first two tests. What score on the last test will give you an A? Question: What test score will make the total 270 or more? UNDERSTAND the problem: Data: The first two test scores are 91 and 86. Develop and carry out a PLAN: Let x be your score on the last test. Clarifying the question Identifying the given data Using a variable to represent what your are trying to find. Total score ≥ 270 Translating to an inequality x ≥ 270 Solving the inequality x ≥ 270 x ≥ 93

CHECK YOUR ANSWER If the third score is 93, = 270 If the third score is greater than 93, say 95, = 272. The score on the third test must be at least 93 for you to receive an A. Replacing x with 93 The answer makes sense in the problem Find the ANSWER and CHECK.

TRY THIS…. In a chemistry course with five tests, a total of 400 points is needed to get a B. Your first four test scores are 91, 86, 73, and 79. What must you score on the last test to get a B? Solve:

MORE EXAMPLES On your new job you can be paid in one of two ways: Plan A: A salary of $1600 per month plus a commission of 4% of total sales. Plan B: A salary of $1800 per month plus a commission of 6% of total sales over $10,000. For what amount of total sales is Plan A better than Plan B, assuming that total sales are always more than $10,000?

SOLUTION Question: What total sales will make the pay for Plan A greater than the pay for Plan B? UNDERSTAND the problem: Data: Plan A: $1600/month salary plus 4% commission Clarifying the question Identifying the given data Plan B: $1800/month salary plus 6% commission on sales over $10,000

SOLUTION Develop and carry out a PLAN: Use x to represent the sales for the month Writing an expression for the income from each plan Income from Plan B = (x – 10,000)0.06 Translating to an inequality Income from Plan A = %x Solving the inequality Income from Plan a > income from Plan B x > (x – 10,000) x > x 400 > 0.02x 20,000 > x

CHECK YOUR ANSWER For the total sales between $10,000 and $20,000, Plan A is better than Plan B. The answer makes sense in the problem Find the ANSWER and CHECK.

TRY THIS…. A painter can be paid in two ways: Plan A: $500 plus $45 per hour. Plan B: $55 per hour Suppose the job takes n hours, For what values of n is Plan A better for the painter? Solve: