Absolute Value Equations


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1 Absolute Value Equations Discussion: Absolute value refers to the measure of distance from zero f any value on the number line. F example, the absolute value of 3 is 3 (written as ) because there are three units between the number 3 and zero on the number line. Negative 3 (3) is also three units from 0, so its absolute value is also 3 (written as.) If I said to you, the absolute value of some unknown (call it x) is 3, stated mathematically as would you conclude? In other wds, what does x equal?, what Let s wk on a problem to see these ideas at wk. If we know that 24 is the absolute value of the number that is 4 less than x, what are the possible values f x? First step is to restate the English sentence into a mathematical one. It s pretty easy to see that in the equation above x might equal 28. But isn t there a second answer here? The absolute value bars are neutralizing a possible negative sign on the side containing the absolute value bars so this problem actually must be solved twice. One way: Second way: Graphically we would represent this as shown below: General Rule: If the value of the absolute is greater than 0, there must be two solutions. Do we have the same dilemma with this problem? NO! Zero is never negative. =4 General Rule: If the value of the absolute is equal to 0, there is only one solution.
2 What about this one? EEk! If the value of the absolute is the measure of the distance to zero, it can never be a negative number (we don t count with negative numbers do we?) so there are no real solutions to this problem. General Rule: If the value of the absolute is less than 0, there are no real number solutions. To Summarize these ideas: Practice: Solve the following absolute value problems f all possible solutions. 1) 2) 3) 4) 5) 6)
3 Absolute Value Inequalities: In equations, we find exact values f the unknown; means x is exactly 2 units away from zero. On the other hand, inequalities express a range of values. An example from yesterday, has as its solutions all number greater than 2. We, also, found that we can represent this graphically. On the line below, graph. If the inequality involves an absolute value, the sentence changes dramatically. The expression states, The distance from x to 0 is greater than 2 units on both sides of zero. Stated another way, if the absolute value of x is greater than two then x is me than 2 less than 2 because both numbers have an absolute value of 2. See this represented graphically below Now, let s wk this out mathematically to convince ourselves this is a true statement. (divided both sides by 1) Remember, the absolute value bars hide the fact that the value of the absolute might actually be negative. We have to take this into account by solving twice, once f a possible positive value on the left and once f the possible negative value. Here s a me complex example to reinfce what we ve talked about up to now. Graph the solution we got fm the above problem But just when you think you ve got this thing figured out, they throw another twist at you. Consider this mathematical statement. The expression states, The distance from x to 0 is less than equal to 2 on both sides of zero. Stated another way, if the absolute value of x is less than equal to 2 then x is
4 the region between 2 and 2. (Note: since this is a less than equal to symbol both end points are included.) Again, let s wk this out mathematically to convince ourselves this is a true statement. (dividing both sides by negative 1) However, we can state this me efficiently using a compound statement that looks just like the number line. In fact, you should always restate this type of inequality as a compound statement. Reading this mathematical expression from left to right we see that negative 2 is less than equal to the middle relation, x. Actually there is a better way to do this. You can always rewrite the iginal absolute value problem of this type as a compound inequality and solve it that way. Your answer will then automatically be in its crect fm. Let s do one like this: (Rewrite as a compound inequality) To review these ideas consider the following table: (Add three to all parts to isolate x) Sentence Meaning Solution The distance from x to 0 is exactly a units The distance from x to 0 is greater than a units. The distance from x to 0 is less than a units Practice: Solve the following absolute value problems f all possible solutions and draw the related graphs. 7) 8)
5 9) 10) 11) 12) F #13, #14, and #15 choose a variable and write as an absolute value inequality that represents the set of numbers on a number line. 13) All numbers no me than six units from zero. 14) All numbers at least 515 units from zero. 15) x is within 5 units of 2 Using Absolute Value to Solve Radical Inequalities In R2 we found that if x is positive zero,. But what if x is negative? Let s look at an example:. From this we see that f negative x,. So f any real number,. But this is exactly how we defined, so f any real number x,.
6 This is the same as solving: So, is our solution. Practice: Solve the inequality. Write your answers in both inequality and interval notation. 16)
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