Solving Basic Square Root Inequalities
Description/Explanation/Highlights
Video Description
This video explains how to solve basic square root inequalities.
Steps and Key Points to Remember
To solve a square root inequality, follow these steps:
- Square root inequalities are solved in the same way as equations except when multiplying or dividing by a negative number as part of finding the solution, the inequality sign must be flipped.
- In addition, when solving a square root inequality, we must first take into account limits that may exist to the domain that would cause a number under the square root to be negative.
- To find limits to the domain, set the portion of the equation under the square root sign \(\geq 0\) and solve the inequality. Any answer that you get from solving the entire inequality may be limited by this domain restriction.
- To solve a square root inequality:
- Isolate the square root.
- Square both sides to get rid of the square root.
- Solve the resulting inequality remembering to flip the inequality sign if division or multiplication by a negative number is required.
Compare the solution set to the limits to the domain you found above to see if the solution must be limited.
- To solve the square root inequality \(\sqrt{x+7}\geq 3\), first find limits to the domain by setting the part under the square root sign, \(x+7\geq 0\).
- Solving this inequality by subtracting 7 from both signs results in the solution set \(x\geq -7\). All answers in the final solution set must be greater than or equal to -7.
- Solve the inequality\(\sqrt{x+7}\geq 3\), by first isolating the square root if needed. The square root in this problem is already isolated.
- Now square both sides. This will get rid of the square root sign on the left and square 3 leaving, \(x+7\geq9\).
- Solve the simple linear inequality and the solution is \(x\geq2\).
- Compare the solution set to the domain limits above. All numbers that are greater than or equal to 2 are also greater than or equal to -7 so all of the numbers in the solution set are acceptable. The final solution is \(x\geq2\).
- The inequality \(\sqrt{x-4}\leq5\) is different, however, and will result in a restricted domain limiting the solution set.
- Find the domain restrictions by setting the portion under the square root sign \(\geq0\) as before and solving. \(x-4\geq0\longrightarrow x\geq4\).
- Solve the original equation by squaring both sides (since the square root is already isolated) and adding 4. The solution will be \(x\leq29\).
- Because of the domain restriction, only numbers greater than or equal to 4 are allowed and there are many values of \(x\leq29\) that are not greater than 4 so we must restrict the solution set to only those numbers less than or equal to 29 but also greater than or equal to 4. In other words the numbers between 4 and 29 including 4 and 29.
- This solution set is written \(4\leq x\leq 29\) or in interval notation \([4,29]\).
Here are some key points to keep in mind when solving basic square root inequalities.
- Square root inequalities are solved in the same basic way as square root equations except the inequality sign must be flipped if multiplying or dividing by a negative number is involved in the solution.
- Domain restrictions must be determined before finding the final solution.
- To find domain restrictions, set the portion of the equation under the square root \(\geq0\) and solve.
- Compare the final result of solving the equation to the domain restrictions to see if any part of the solution set must be eliminated before writing the final answer.
Video Highlights
- 00:00 Introduction
- 00:55 \(\sqrt{x+7}\geq3\) example of solving a square root inequality
- 04:08 \(\sqrt{x-4}\leq5\) example of solving a square root inequality with a restricted domain
- 06:24 Conclusion
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