Solving Inequalities

Example 1 These are called inequalities because they are like equations, except they don’t have an equals sign. They still follow most of the same rules. Work the problem out as if it has an equals sign. Then when you get an answer, sketch the graph on the number line and write your answer in interval notation.

Solving Inequalities Example 2 Why is this different from Example 1? Because it has an x on both sides of the inequality. However, it doesn’t change how you work the problem.

Solving Inequalities Example 3 This is the first example of a problem that is a little different than solving equations. Anytime you multiply or divide by a negative, you have to turn the inequality symbol around.

Solving Inequalities Example 4 For this problem, since the variables subtracted off, it’s either “no solution” or “all real numbers”. It is “no solution” if the statement leftover is false. It is “all real numbers” if the statement is true.

Solving Inequalities Example 5 Is this “no solution” or “all real numbers”? Neither. It’s only one of those if the variables go away.

Solving Rational Equations Example 6 Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left. ( ) 10

Solving Inequalities Practice: Answers:

Solving Absolute Value Equations

Example 1 This problem has absolute value bars in it. Anytime you see absolute value bars in an equation, you need to split the problem into two different problems. The first equation is the exact as the original except just erase the absolute value bars. For the second equation, just change the sign of the other side. OR

Solving Absolute Value Equations Example 2 OR

Solving Absolute Value Equations Example 3 Always make sure that the absolute value bars are alone first, so add the four to both sides before you split it into two. OR

Solving Absolute Value Equations Example 3.5 You cannot distribute numbers into the absolute values. Since the negative three is being multiplied times the absolute value bars, to get rid of them, we need to divide both sides by the negative three. OR

Solving Absolute Value Equations Example 4 Because the absolute values can never equal a negative, there is no work involved on this problem.

Solving Absolute Value Equations Example 5 Whenever there are variables inside the absolute value bars AND outside, you HAVE TO CHECK YOUR ANSWERS!!! OR

Solving Absolute Value Equations Example 5 Remember the rules for checking. 1)Always go back to the original problem. 2)Do not cross the equals sign. 3)Use order of operations on each side. Hurray! Both of them worked.

Solving Absolute Value Equations Example 6 Do I have to check my answers here? OR

Solving Absolute Value Equations Example 6 Since this gives you a true statement, 4 works. Therefore, x = 4.

Solving Absolute Value Equations Example 6 How do you communicate that? Mark off the answers that don’t work, and circle the ones that do. What is your answer if NEITHER answer checks? OR

Solving Absolute Value Equations Example 7 OR

Solving Absolute Value Equations Example 7 Since neither one works, there are NO SOLUTIONS.

Solving Absolute Value Inequalities and Compound Inequalities

Solving Absolute Value Inequalities Example 6 Because there is an absolute value in the problem, that tells me that I have to split the problem into two pieces. When you write it the second time, not only do you change the sign, but you also turn the inequality around. To decide if you use “and” or “or”, remember GO to LA. Greater than Or Less than And With “or”, just put both inequalities on the final graph.

Solving Absolute Value Inequalities Example 7 Because there is an absolute value in the problem, that tells me that I have to split the problem into two pieces. To decide if you use “and” or “or”, remember GO to LA. Greater than Or Less than And When you write it the second time, not only do you change the sign, but you also turn the inequality around. With “and”, find where the two inequalities intersect, and put that on the final graph.

Solving Absolute Value Inequalities Example 8

Solving Absolute Value Inequalities Example 9 When there is a negative on the other side of an absolute value inequality, the answer is either “no solution” or “all real numbers”. Because the absolute value will always be positive, if it is a greater than, it will be “all real numbers”. If there is a less than sign with the negative on the outside, the answer is “no solution”. Example 10

Solving Absolute Value Inequalities Example 11

Solving Inequalities Example 12 YOU DO NOT BREAK THIS INTO TWO PROBLEMS BECAUSE THERE ARE NO ABSOLUTE VALUE BARS!!!

Solving Absolute Value Inequalities Example 13 This is a compound inequality. It is already set up to start solving the separate equations. Since it has an “or” between the two, just put both graphs on the final graph and write your answer in interval notation.

Solving Absolute Value Inequalities Example 14 This is another type of compound inequality. Whatever you do to get the x by itself in the middle, you have to do it to all “sides” of the inequality. Since it is written with two inequalities in one sentence, it is understood to have an “and” between them. Therefore, solve, and find the intersection.

Solving Absolute Value Inequalities Practice.Answers:

Solving Absolute Value Inequalities Your turn. Do the worksheet on Inequalities. You may work together. This is for homework if you do not finish during class. Show your work!!!

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