Tags:How to Solve Absolute Value Inequalities,absolute value inequalities
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In this video we are going to talk about solving absolute value and inequalities. Before we move to absolute value inequalities, let’s go ahead and refresh ourselves with absolute value equations. This equation says that a number of distance for a zero is three and two numbers fit that at both – +/-3. So if you really look at that on a graph, it looks something like that. Now if we want to think about it as inequality. This no longer says a number whose distance is three. This is a set of numbers whose distances are less than or equal to three.
Again that is set of numbers whose distances from zero are less than or equal to three. So if you want to look at it on a graph it’s all the numbers in between -3 and 3 inclusive with the closed set and close interval, this inequality right here says “the set of numbers who’s distance away from zero is bigger than three. Greater than three so that is from three all the way to positive infinity and from -3, all the way to negative infinity. Now let’s go ahead and I think the way to look it solving an absolute value inequality.
So this says this quantity which is represented by 3a – 1 has a distance away from zero less than five. So we are going to put five in there, get it 5 on the number line and we are going to stick 3a – 1 as the quantity whose distance is in between the two. Now if we add one to everybody, you now have 3a is going to be in between -5 +1 is -4. And 5 +1 is 6, then let’s divide by three and a is going to be in between -4/3 and two. We can write that as a is which is less than or equal to two and greater than or equal 4/3.
Now let us take a look at the actual inequality and solve it algebraically. We are going to write this as two separate inequalities and we are going to move the absolute value bars. We are going to use the right hand side and just re-write it as 3a – 1 < 5. And now we are going to switch the inequality symbol and switch to sign of the number and we are going to have 3a – 1 is >-5.
We are going to solve each separately, add one to both sides, we got 3a < 6 divided by three and we have a < 2. Now we are going to do this one, we are going to add one to both sides and get 3a > -4 and dividing by three. We have a > -4/3 so we have a < 2 a >-4/3.
So now let’s go ahead and take a look at another time of inequality, let’s take a look at how about a very complicated one that you think might have to be a lot of work. Three times the absolute value 2x – 2 +2 >/= -5. So let’s go ahead and before we go ahead and solve this out in this inequality, we’re going to clear away the positive two and the three then we can go ahead and write it as two separate inequalities like we did up here. Let’s go and subtract two from both sides and we will let the three times the absolute value of 2x – 2 >/=-7. And then divide by three and we have the absolute value 2x – 2 >/= -7/3.
Now stop here because you don’t have to do anymore, remember absolute value is a distance. And we are talking about a distance from zero. Every number on the real number line has a distance away from zero that is greater than zero or zero so any number in these bars will work. So this solution is all real numbers. You don have to do anymore because distances are always bigger than zero or zero themselves.
Now let’s take a look at one more type of problem. Let’s try the absolute value of -5 – a > 4. So there is nothing to clear away from outside the absolute value bars, so now we are just going to split it and write it as two separate inequalities, -5 – a >4. And -5 – a < -4, so we are going to add 5 to both sides and we are going to have a > 9 and then divide by -1 and if you recall whenever you multiply divided by -1 you switch to direction of inequality symbol.
Here we are going to add 5 and we have -a < -4 +5 I = 1 and divide by the negative. We are going to have a > -1. So, on the number line if we were asked to graph it. we would need -9. We would need -1 and put zero there for our reference, leave all the numbers that are less -9. So that goes up that way and all the numbers that are greater than -1 which goes out that way.
There are to cover for absolute value and inequalities, grab and rewind this thing and watch as many times as you need to until you get tired of my voice.