Solving Rational Inequalities (2.13.1)

Question 1

• A rational inequality is an inequality that contains one or more rational expressions.

Answer 1

• A rational inequality is an inequality that contains one or more rational expressions.

Question 2

• The solution for a rational inequality will be at least one interval of x-values (i.e., region on a number line), bounded by
the rational inequality’s critical values. A rational inequality’s critical values are the x-values that make the rational inequality
either equal to 0 or undefined.

Answer 2

• The solution for a rational inequality will be at least one interval of x-values (i.e., region on a number line), bounded by
the rational inequality’s critical values. A rational inequality’s critical values are the x-values that make the rational inequality
either equal to 0 or undefined.

Question 3

• Steps for Solving a Rational Inequality 1. Set the rational expression in relation to 0. 2. Find the critical values. A. Set the numerator equal to 0 and solve to find the x-value where the inequality is equal to 0. B. Set the denominator equal to 0 and solve to find the x-value where the inequality is undefined. 3. Mark the critical values on a number line, dividing it into regions. 4. Choose a random value from one of these regions on the number line and test that value in the inequality (substitute the value into the inequality for x and simplify). If the result is true, then that entire region is part of the inequality’s solution. Repeat for each region on the number line. 5. Write the solution using inequalities or interval notation.

Answer 3

• Steps for Solving a Rational Inequality 1. Set the rational expression in relation to 0. 2. Find the critical values. A. Set the numerator equal to 0 and solve to find the x-value where the inequality is equal to 0. B. Set the denominator equal to 0 and solve to find the x-value where the inequality is undefined. 3. Mark the critical values on a number line, dividing it into regions. 4. Choose a random value from one of these regions on the number line and test that value in the inequality
(substitute the value into the inequality for x and simplify). If the result is true, then that entire region is part of the inequality’s solution. Repeat for each region on the number line. 5. Write the solution using inequalities or interval notation.

Question 4

In this example, a rational inequality is given, along with a
possible solution, and the question is, “Is 2 a solution to the
inequality?” To determine if 2 is a solution, substitute 2 into
the inequality for x and simplify. If the result is a true
statement, then 2 is a solution. If the result is a false statement,
then 2 is not a solution. When 2 is substituted into the
inequality, the result is 15 > 0. Since this is a true statement,
2 must be a solution to the inequality. Note that 2 is not the
only solution. The solution for this inequality is actually a
range of values.

Answer 4

In this example, a rational inequality is given, along with a
possible solution, and the question is, “Is 2 a solution to the
inequality?” To determine if 2 is a solution, substitute 2 into
the inequality for x and simplify. If the result is a true
statement, then 2 is a solution. If the result is a false statement,
then 2 is not a solution. When 2 is substituted into the
inequality, the result is 15 > 0. Since this is a true statement,
2 must be a solution to the inequality. Note that 2 is not the
only solution. The solution for this inequality is actually a
range of values.

Question 5

Solutions to this inequality are all values of x that make the rational expression negative. To determine if −3 is a solution to the given inequality, substitute −3 into the inequality for x and simplify. If the result is a negative number, then −3 is a solution. The result of substituting and simplifying is 0. Since 0 is not a negative number, −3 is not a solution to this inequality.

Answer 5

Solutions to this inequality are all values of x that make the rational expression negative. To determine if −3 is a solution to the given inequality, substitute −3 into the inequality for x and simplify. If the result is a negative number, then −3 is a solution. The result of substituting and simplifying is 0. Since 0 is not a negative number, −3 is not a solution to this inequality.

Question 6

When solving a rational inequality, the first step is to write
the inequality with 0 on one side. In this example, the right
side is already 0, so step 1 can be skipped.
Step 2 is to find the critical values. The first critical value is
the value of x that makes the rational expression equal to 0.
A rational expression is equal to 0 when the numerator is 0.
Set the expression in the numerator, x − 2, equal to 0 and
solve to find this value.
The second critical value is the value of x that makes the
rational expression undefined. A rational expression is
undefined when the denominator is 0. Set the expression in
the denominator, x + 3, equal to 0 and solve to find this value.
The critical values divide the number line into regions. The
solution for a rational inequality is one or more of these
regions. To determine which region is part of the solution,
test any value from that region in the inequality.
Here, the first region is x 2.
The given inequality in this example is not set in relation to
0. So, the first step is to manipulate the inequality so that one
side (usually the right side) is 0. Since the right side is 2, just
subtract 2 from each side to get 0 on the right side.
After the rational expression is set in relation to 0, find the
critical values by setting the expression in the numerator and
the expression in the denominator equal to 0 and solving.
Remember, the critical value found when the denominator is 0 is never included in the solution region.

Answer 6

When solving a rational inequality, the first step is to write
the inequality with 0 on one side. In this example, the right
side is already 0, so step 1 can be skipped.
Step 2 is to find the critical values. The first critical value is
the value of x that makes the rational expression equal to 0.
A rational expression is equal to 0 when the numerator is 0.
Set the expression in the numerator, x − 2, equal to 0 and
solve to find this value.
The second critical value is the value of x that makes the
rational expression undefined. A rational expression is
undefined when the denominator is 0. Set the expression in
the denominator, x + 3, equal to 0 and solve to find this value.
The critical values divide the number line into regions. The
solution for a rational inequality is one or more of these
regions. To determine which region is part of the solution,
test any value from that region in the inequality.
Here, the first region is x 2.
The given inequality in this example is not set in relation to
0. So, the first step is to manipulate the inequality so that one
side (usually the right side) is 0. Since the right side is 2, just
subtract 2 from each side to get 0 on the right side.
After the rational expression is set in relation to 0, find the
critical values by setting the expression in the numerator and
the expression in the denominator equal to 0 and solving.
Remember, the critical value found when the denominator is 0 is never included in the solution region.

Question 7

The inequality symbol in this example includes “or equal to.”
So, the value of x that makes the rational expression equal to
0 is in the solution. This rational expression is equal to 0
when x = −3. So, −3 is in the solution and a closed circle
appears at −3 on the number line. The open circle at −1 is
used to indicate that −1 is not in the solution.
The solution region is the values of x between −3 and −1,
including −3 but not including −1. To write this solution in
interval notation, use a bracket before −3 and a parenthesis
after −1. The bracket indicates that the adjacent value is
included in the solution and the parenthesis indicates that the
adjacent value is not in the solution.
Start the solving process by getting 0 on the right side. Be
sure to get a common denominator when adding and
subtracting with rational expressions. Here, the common
denominator is 2x + 1.
Now find the critical values by setting the expression in the
numerator and in the denominator equal to 0.
Note that since the inequality symbol does not include “or
equal to,” the critical value where the rational expression is
equal to 0 will not be included in the solution. The critical
value where the rational expression is undefined is never
included in the solution. Since neither critical value is
included in the solution, open circles are used at both values
on the number line.
Remember, when testing a value from a region to determine
if it is part of the solution, any value from that region can be
used. So, choose values that are easy to work with.
The solution for this inequality is two regions. Therefore, use
the union symbol when writing the solution in interval
notation. The first solution region is x 2/3. Since 2/3 is not in the
solution, use an opening parenthesis for this interval.

Answer 7

The inequality symbol in this example includes “or equal to.”
So, the value of x that makes the rational expression equal to
0 is in the solution. This rational expression is equal to 0
when x = −3. So, −3 is in the solution and a closed circle
appears at −3 on the number line. The open circle at −1 is
used to indicate that −1 is not in the solution.
The solution region is the values of x between −3 and −1,
including −3 but not including −1. To write this solution in
interval notation, use a bracket before −3 and a parenthesis
after −1. The bracket indicates that the adjacent value is
included in the solution and the parenthesis indicates that the
adjacent value is not in the solution.
Start the solving process by getting 0 on the right side. Be
sure to get a common denominator when adding and
subtracting with rational expressions. Here, the common
denominator is 2x + 1.
Now find the critical values by setting the expression in the
numerator and in the denominator equal to 0.
Note that since the inequality symbol does not include “or
equal to,” the critical value where the rational expression is
equal to 0 will not be included in the solution. The critical
value where the rational expression is undefined is never
included in the solution. Since neither critical value is
included in the solution, open circles are used at both values
on the number line.
Remember, when testing a value from a region to determine
if it is part of the solution, any value from that region can be
used. So, choose values that are easy to work with.
The solution for this inequality is two regions. Therefore, use
the union symbol when writing the solution in interval
notation. The first solution region is x 2/3. Since 2/3 is not in the
solution, use an opening parenthesis for this interval.