3. All the values of satisfying the inequality
are
Correct Answer :
or
Solution :
The correct option is:
or
Step-by-Step Explanation:
We are given the rational inequality:
Below, we analyze this inequality step-by-step, referencing the method displayed in the provided image.
Step 1: Finding the Domain Constraints
The denominators of the fractions must not be equal to zero. As shown in Step 1 of the image:
and
Solving these constraints gives:
and
Step 2: Sign Analysis of the Denominators
To safely solve or cross-multiply, we must determine the sign of the denominators. In Step 2 of the image, the product of the two linear terms is analyzed under:
Expanding this quadratic expression:
Simplifying:
Step 3: Finding the Roots of the Quadratic Equation
As shown in Step 3 of the image, we solve the quadratic equation:
Using the quadratic formula:
This gives the two roots:
Step 4: Testing Intervals
As specified in Step 4 of the image, we test the signs of the quadratic expression in the intervals determined by these roots:
1) For
:
both factors
and
are negative, meaning their product is positive:
.
2) For
:
the factor
is positive and
is negative, meaning their product is negative:
.
3) For
:
both factors
and
are positive, meaning their product is positive:
.
Therefore, the inequality holds for the intervals where the denominators share the same sign:
or
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