Let's look at solving equations and inequalities involving absolute value.
Our first example is an equation.
So we have an expression inside the absolute value,
that is equal to a positive real number.
There are two possibilities.
Either that expression, 4x plus 5, equals positive 3 or it must equal negative 3.
We need to solve each of these linear equations to find our solutions.
Subtract 5 from both sides.
So 3 minus 5 will be negative 2. Divide by 4.
Negative 1/2 is one value of x that satisfies the equation.
Subtract 5 from both sides of the second one. We get negative 8.
We are going to divide both sides by 4 again.
So now we have two solutions to the equation.
x could be negative 1/2 and negative 2. Those are our solutions.
We can write this in set notation using braces.
Now let’s look as some inequalities.
Here is our first absolute value inequality.
So we have the expression x minus 3. It's absolute value
is less than or equal to the real number 1.
So we know that x minus 3 needs to be something from negative 1 to 1.
Now I have a compound inequality and I can solve it,
by getting the x isolated in the middle here.
Add 3 to each part.
So now I have the inequality with the x isolated,
and all I need to do is write the answer in interval notation.
Notice that my end points are included since it was a less than or equal to inequality.
So I am going to use square brackets on my interval.
Let’s look at a second example.
Here is our third example.
This is another absolute value inequality.
We have the absolute value of 2x plus 1 is greater than this positive number 5.
There are two possibilities.
2x plus 1 is greater than 5 or
the other possibility is that 2x plus 1 is less than negative 5.
Leaving this off is the most common mistake students make in solving these problems.
So again we have two linear inequalities. We need to solve each one.
Subtract 1 from both sides here. Divide by 2. This is part of our solution.
Now let’s solve the other one. Subtract 1 from both sides. Divide by 2.
Alright, so our solution is all the real numbers x that are bigger than 2
or less than negative 3.
And I need to write this in interval notation.
Now it is customary to write the interval with the smaller numbers first,
on the left, so let’s start with this one.
This is all real numbers less than negative 3.
I do not want to include that negative 3, so use parentheses.
And I want anything that is smaller than negative 3,
so that is towards negative infinity, and that is also never included.
There is the first one.
The other interval is all real numbers x that are bigger than 2.
So these will be all the numbers from 2 to infinity.
Again, parentheses to show the end points are not included.
So the solution is all the real numbers that are in this interval or in this one.
And a mathematical symbol we could there is the union symbol.
Now be sure to practice solving these types of inequalities and equations
until you are comfortable with the process.