Absolute Value Inequalities and Equations
By ashbybellows, Updated
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Not fair! Mommy said I can't have all the candy in the store because I only have five dollars! I need help!!
Aahh! What just happened?
Fear not! For I, Sir Absolute Value Equations and Inequalities am here to help. (You can can me SAVEAI for short.)
Step 1: Distribute the -6 Step 2: You now have 3| X-7 | > 6; distribute the 3 by dividing both sides of the equation by 3 Step 3: You now have |X-7| > 2. Next create two separate equations, one with a >2 and one with a <-2. Step 4: You now have X-7 >2 and X-7<-2; next simplify these inequalities. Step 5: You now have X>9 and X<5; this is not your final answer and you must first graph it by using either a closed dot (includes the number, equal to) or an open dot (does not include the number) on a number line. Here is the graph for this problem: Step 6: The graph indicates what type of solution it is. This problem it is an or question because you cannot have a number greater than 9 and less than 5. You can only have one or the other. Therefore the final solution is X>9 or X<5.
I am going to teach you how to solve these simple equations and inequalities which can than help you figure out how much candy you can get for five dollars.
Inequality One: 3I X-7 I -6 > 0
I'm confused! Why did you make two equations earlier with a 2 and -2?
Now let's apply what we've learned to solving absolute value equations. The principle is the same, in fact the only difference is that there is an equal sign not an inequality.
Equation One: |x + 3| -7 = 2 Step 1: Distribute the -7 Step 2: You now have |x + 3|= 9; next set up two equations x + 3 = 9 and x + 3= -9 Step 3: Solve the equations Step 4: The answer is: x= 6 and x= -12
Ah. So the two equations I made earlier were actually a shortcut to make the math you are already are doing shorter. ABSOLUTE VALUE can be defined as the distance between a number and zero. Therefore if you are trying to solve |x|=16 the answer can either be x= 16 or x= -16 because -16 is still 16 away from zero. An easy way to think about it is a number inside of absolute value will always be positive because you cannot be a negative distance away from zero.
Okie dokie! I have five dollars for candy and I want to get as much candy as possible. I know whatever I get has to be less then or equal to 5 dollars, and I have a 50% discount. Each candy is worth one dollar. Therefore I can set up the equation like this: |x|< 5 where x is the amount of candy because the amount is always a distance away from zero. Inequality Two: |x|< 5 Step 1: Distribute the 2 by multiplying both sides by 2 Step 2: You now have |x|<10 or x <10 solve by creating the second inequality x>-10 Step 3: After graphing: The answer is -10<x<10 This means that if each candy costs one dollar I can by a minimum of 0 candies and a max of 9 candies.
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