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Answer
The solution set is
Solution
Solve the equation
Solve the inequality by following steps:
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
- step1: Find the domain:
- step2: Move the expression to the left side:
- step3: Subtract the terms:
- step4: Change the sign:
- step5: Set the numerator and denominator of
equal to 0
- step6: Calculate:
- step7: Determine the test intervals:
- step8: Choose a value:
- step9: Test the chosen value:
- step10: Include the critical value:
- step11: The final solution is
- step12: Check if the solution is in the defined range:
- step13: Find the intersection:
The solution to the inequalityis .
Therefore, the solution set in interval notation is
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Simplify this solution The Deep Dive
To solve the inequality
-
Subtract 3 from both sides to get
. -
Combine the terms:Simplify the numerator:This leads us to:
-
Now, find the critical points where the expression is equal to zero or undefined:
- The numerator
gives . - The denominator
gives .
- The numerator
-
Perform a sign analysis on the intervals
, , and : - For
, both the numerator and denominator are negative, leading to a positive value. - For
, the numerator is negative and the denominator is positive, resulting in a negative value. - For
, both the numerator and denominator are positive, giving a positive value.
- For
-
The inequality
holds for and . We do not include since it makes the denominator zero.
Thus, in interval notation, the solution set is
The solution set is
While inequality solving can feel dry, it’s been a fundamental part of mathematics for centuries, affecting everything from economics to engineering! Fun fact: historical mathematicians, like Descartes, laid down the groundwork for modern algebraic inequalities, leading to our current practices.
In the real world, understanding inequalities is crucial when you’re budgeting or planning. For instance, if you analyze your monthly expenses and set limits, you’re essentially solving inequalities! So next time you calculate whether you can buy that new gadget within your budget, think of it as a fun math challenge!
