Pregunta
Solve the following inequality algebraically
A. The solution does not have any isolated values. It can be described as
(Type your answer in interval notation)
B. The solution consists exclusively of one or more isolated values It can be described as {
(Use a comma to separate answers as needed.)
C. The solution has at least one isolated value It can be described as un }.
(In the first answer box, type your answer in interval notation. In the second answer box, ust
separate answers as needed.)
A. The solution does not have any isolated values. It can be described as
(Type your answer in interval notation)
B. The solution consists exclusively of one or more isolated values It can be described as {
(Use a comma to separate answers as needed.)
C. The solution has at least one isolated value It can be described as un }.
(In the first answer box, type your answer in interval notation. In the second answer box, ust
separate answers as needed.)
Ask by Davey Osborne. in the United States
Jan 17,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The solution is all real numbers less than -7, between -3 and -2, 0, and greater than 3.
Solución
To solve the inequality
we need to determine where the expression is nonnegative.
Step 1: Identify Critical Points
The critical points occur where the numerator or denominator is zero:
The critical points occur where the numerator or denominator is zero:
- Numerator:
(double root), , . - Denominator:
, (these points are excluded as they make the denominator zero).
Step 2: Determine Intervals
The critical points divide the real number line into the following intervals:
The critical points divide the real number line into the following intervals:
Step 3: Test Each Interval
Evaluate the sign of the expression in each interval:
Evaluate the sign of the expression in each interval:
-
: - Numerator: Positive.
- Denominator: Positive.
- Expression: Positive.
-
: - Numerator: Negative.
- Denominator: Negative.
- Expression: Positive.
-
: - Numerator: Positive.
- Denominator: Negative.
- Expression: Negative.
-
: - Numerator: Negative.
- Denominator: Negative.
- Expression: Positive.
-
: - Numerator: Positive.
- Denominator: Negative.
- Expression: Negative.
-
: - Numerator: Positive.
- Denominator: Positive.
- Expression: Positive.
Step 4: Include Boundary Points
-
and make the numerator zero, so they are included. -
makes the numerator zero, so it is included. -
and are excluded as they make the denominator zero.
Final Solution:
The solution consists of intervals where the expression is nonnegative and includes the isolated point
The solution consists of intervals where the expression is nonnegative and includes the isolated point
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Simplifique esta solución The Deep Dive
To solve the given inequality
we first need to determine the critical points by setting the numerator and denominator to zero.
The numerator
The denominator
Now, we identify our critical points:
Next, we analyze the sign of the expression in each of the intervals defined by these critical points:
Calculating the sign of the expression in each interval:
- For
(e.g., ): Positive - For
(e.g., ): Negative - For
(e.g., ): Positive - For
(e.g., ): Negative - For
(e.g., ): Positive - For
(e.g., ): Positive
Now, we summarize the sign changes:
- The expression is non-negative (≥ 0) in the intervals:
-
Next, let’s note the critical points:
-
makes the denominator zero (not included). - The numerator is zero at
, , and (included). - We can include
in a non-strict inequality (can’t include it since it’s in the denominator).
Thus, the solution in interval notation is:
The intervals that satisfy the inequality are:
Therefore, the answer is:
A. The solution does not have any isolated values. It can be described as:
A. The solution does not have any isolated values. It can be described as:
In interval notation, this shows the continuous ranges without single isolated values.
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