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Example Questions
Example Question #1 : Linear Inequalities
Solve and graph:
1) Write as two simple inequalities:
2) Solve the inequalities:
3) Write the final solution as a single compound inequality:
For interval notation:
4) Now graph:
Example Question #2 : Linear Inequalities
Solve for
In order to solve this equation, we must first isolate the absolute value. In this case, we do it by dividing both sides by which leaves us with:
When we work with absolute value equations, we're actually solving two equations. So, our next step is to set up these two equations:
and
In both cases we solve for by adding
to both sides, leaving us with
and
This can be rewritten as
Example Question #3 : Linear Inequalities
Solve for
When we work with absolute value equations, we're actually solving two equations:
and
Adding to both sides leaves us with:
and
Dividing by in order to solve for
allows us to reach our solution:
and
Which can be rewritten as:
Example Question #1 : Linear Inequalities
Solve for
In order to solve for we must first isolate the absolute value. In this case, we do it by dividing both sides by 2:
As with every absolute value problem, we set up our two equations:
and
We isolate by adding
to both sides:
and
Finally, we divide by :
and
Example Question #5 : Linear Inequalities
Solve for .
Our first step in solving this equation is to isolate the absolute value. We do this by dividing both sides by
.
We then set up our two equations:
and
.
Subtracting 4 from both sides leaves us with
and
.
Lastly, we multiply both sides by 2, leaving us with :
and
.
Which can be rewritten as:
Example Question #1 : Solve Absolute Value Inequalities
Solve for
We first need to isolate the absolute value, which we can do in two steps:
1. Add 2 to both sides:
2. Divide both sides by 4:
Our next step is to set up our two equations:
and
We can now solve the equations for by subtracting both sides by 8:
and
and then dividing them by 5:
and
Which can be rewritten as:
Example Question #1 : Solve Absolute Value Inequalities
Solve the following absolute value inequality:
First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting seven from both sides.
Next we need to set up two inequalities since the absolute value sign will make both a negative value and a positive value positive.
From here, subtract thirteen from both sides and then divide everything by four.
Example Question #8 : Linear Inequalities
Solve the following absolute value inequality:
First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by dividing both sides by three.
We now have two equations:
and
So, our solution is
Example Question #1 : Solve Absolute Value Inequalities
Solve the following inequality:
First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting two from both sides then dividing everything by three.
Since absolute value signs make both negative and positive values positive we need to set up a double inequality.
Now to solve for subtract four from each side.
Example Question #2 : Linear Inequalities
Solve for :
If , then either
or
based on the meaning of the absolute value function. We have to solve for both cases.
a) subtract 5 from both sides
divide by -2, which will flip the direction of the inequality
Even if we didn't know the rule about flipping the inequality, this answer makes sense - for example, , and
.
b) subtract 5 from both sides
divide by -2, once again flipping the direction of the inequality
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