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Example Questions
Example Question #1 : Solving Polynomial And Rational Inequalities
The zeros of the function are the values of where the function will be equal to zero. In order to find these we set the numerator of the function equal to zero.
We only need to solve for once,
So the zeros of this function are .
To solve for the points at which this function will be undefined, we set the denominator equal to zero and solve for .
Example Question #1 : Solving Polynomial And Rational Inequalities
True or False: For the inequality ,
is a potential value for
.
False
True
False
This inequality wants all values where is greater than
. So everything up until
is included and this is represented by having a dotted line on the graph or an open circle on a number line.
Example Question #8 : Solve And Graph Rational Inequalities
Which of the following is an accurate graph of the inequality ?
We will first begin by solving for the zeros and undefined points of the inequality. We solve for the zeros by setting the numerator equal to zero.
And so the zeros of this function are at
Now we will solve for the undefined points by setting the denominator equal to zero. Since the denominator is , then whenever
, the function is undefined. Now we need to find for which values of
is each factor is greater than zero.
For , any value where
will be positive and we will be able to graph it. For
, any value where
will be positive. Now, we can only graph these values up until
because the function is undefined here. We are able to pick up the graph again once we reach
. The graph will look like the one below.
Example Question #1 : Inequalities
Solve the following inequality
No Solution
All Real Numbers
We begin by moving all of our terms to the left side of the inequality.
We then factor.
That means our left side will equal 0 when . However, we also want to know the values when the left side is less than zero. We can do this using test regions. We begin by drawing a number line with our two numbers labeled.
We notice that our two numbers divide our line into three regions. We simply need to try a test value in each region. We begin with our leftmost region by selecting a number less than . We then plug that value into the left side of our inequality to see if the result is positive or negative. Any value (such as
) will give us a positive value.
We then repeat this process with the center region by selecting a value between our two numbers. Any value (such as ) will result in a negative outcome.
Finally we complete the process with the rightmost region by selecting a value larger than . Any value (such as
) will result in a positive value.
We then label our regions accordingly.
Since we want the result to be less than zero, we want the values between our two numbers. However, since our left side can be less than or equal to zero, we can also include the two numbers themselves. We can express this as
Example Question #2 : Inequalities
Solve the quadratic inequality.
We begin by solving the equation for its zeros. This is done by changing the sign into an
sign.
Since we know the zeros of the equation, we can then check the areas around the zeros since we naturally have split up the real line into three sections :
First we check
Therefore, the first interval can be included in our answer. Additionally, we know that satisfies the equation, therefore we can say with certainty that the interval
is part of the answer.
Next we check something in the second interval. Let , then
Therefore the second interval cannot be included in the answer.
Lastly, we check the third interval. Let , then
Which does satisfy the original equation. Therefore the third interval can also be included in the answer. Since we know that satisfies the equation as well, we can include it in the interval as such:
Therefore,
Example Question #4 : Inequalities
Solve:
Method 1:
1) Multiply-out the left side then rewrite the inequality as an equation:
2) Now rewrite as a quadratic equation and solve the equation:
3) Next set up intervals using the solutions and test the original inequality to see where it holds true by using values for on each interval.
4) The interval between and
holds true for the original inequality.
5) Solution:
Method 2:
Using a graphing calculator, find the graph. The function is below the x-axis (less than ) for the x-values
. Using interval notation for
,
.
Method 3:
For the inequality , the variable expression in terms of
is less than
, and an inequality has a range of values that the solution is composed of. This means that each of the solution values for
are strictly between the two solutions of
. 'Between' is for a 'less than' case, 'Outside of' is for a 'greater than' case.
Example Question #1 : Solve And Graph 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 #1 : 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 #2 : 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
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