Absolute Value Equations and Inequalities
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Solve each of the following.
This uses the following fact
The requirement that d be greater than or equal to zero is
simply an acknowledgement that absolute value only returns number that are
greater than or equal to zero. See
Problem 3 below to see what happens when d is negative.
So the solution to this equation
So there were two solutions to
this. That will almost always be the
case. Also, do not get excited about the
fact that these solutions are negative.
This is not a problem. We can
plug negative numbers into an absolute value equation (which is what we’re
doing with these answers), we just can’t get negative numbers out of an
absolute value (which we don’t, we get 2 out of the absolute value in this
This one works identically to the
Do not make the following very
common mistake in solve absolute value equations and inequalities.
Did you catch the mistake? In dropping the absolute value bars I just
changed every “-” into a “+” and we know that doesn’t work that way! By doing this we get a single answer and it’s
incorrect as well. Simply plug it into
the original equation to convince yourself that it’s incorrect.
When first learning to solve
absolute value equations and inequalities people tend to just convert all minus
signs to plus signs and solve. This is
simply incorrect and will almost never get the correct answer. The way to solve absolute value equations is
the way that I’ve shown here.
This question is designed to make
sure you understand absolute values. In
this case we are after the values of x
such that when we plug them into we will get -15. This is a problem however. Recall that absolute value ALWAYS returns a
positive number! In other words, there
is no way that we can get -15 out of this absolute value. Therefore, there are no solutions to this
To solve absolute value
inequalities with < or in them we use
As with absolute value equations
we will require that d be a number that is greater than or equal to zero.
The solution in this case is then
In solving these make sure that
you remember to add the 10 to BOTH sides of the inequality and divide BOTH
sides by the 7. One of the more common
mistakes here is to just add or divide one side.
This one is identical to the
previous problem with one small difference.
Don’t forget that when multiplying
or dividing an inequality by a negative number (-2 in this case) you’ve got to
flip the direction of the inequality.
This problem is designed to show
you how to deal with negative numbers on the other side of the inequality. So, we are looking for x’s which will give us a number (after taking the absolute value of
course) that will be less than -1, but as with Problem 3 this just isn’t possible since absolute value will
always return a positive number or zero neither of which will ever be less than
a negative number. So, there are no
solutions to this inequality.
Absolute value inequalities
involving > and are solved as follows.
Note that you get two separate
inequalities in the solution. That is
the way that it must be. You can NOT put
these together into a single inequality.
Once I get the solution to this problem I’ll show you why that is.
Here is the solution
So the solution to this inequality
will be x’s that are less than -2 or
greater than .
Now, as I mentioned earlier you
CAN NOT write the solution as the following double inequality.
When you write a double inequality
(as we have here) you are saying that x
will be a number that will simultaneously satisfy both parts of the
inequality. In other words, in writing
this I’m saying that x is some number
that is less than -2 and AT THE SAME TIME is greater than . I know of no number for which this is
true. So, this is simply incorrect. Don’t do it.
This is however, a VERY common mistake that students make when solving
this kinds of inequality.
Not much to this solution. Just be careful when you divide by the -11.
This is another problem along the
lines of Problems 3 and 6. However, the
answer this time is VERY different. In
this case we are looking for x’s that
when plugged in the absolute value we will get back an answer that is greater
than -4, but since absolute value only return positive numbers or zero the
result will ALWAYS be greater than any negative number. So, we can plug any x we would like into this absolute value and get a number greater
than -4. So, the solution to this
inequality is all real numbers.