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Chapter 4: Problem 34

Solve. $$11-x^{2} \geq 0$$

Short Answer

Expert verified
-\sqrt{11} \leq x \leq \sqrt{11}.

Step by step solution

01

Rearrange the Inequality

The given inequality is \[11 - x^2 \geq 0.\] Rearrange it to make it easier to solve: \[-x^2 \geq -11.\]
02

Divide by -1

Divide both sides of the inequality by \(-1\), and reverse the direction of the inequality: \[x^2 \leq 11.\]
03

Solve for x

Take the square root of both sides, remembering to include both the positive and negative roots: \[ - \sqrt{11} \leq x \leq \sqrt{11}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality solving
Solving inequalities can sometimes seem tricky, but with a clear process, it becomes much easier.
When solving quadratic inequalities like \( 11 - x^2 \geq0 \), start by isolating the quadratic term. This often involves rearranging the inequality, as we did when we wrote \( -x^2 \geq -11 \).
Remember that when both sides of an inequality are divided or multiplied by a negative number, the inequality sign must be reversed.
In our example, dividing by \(-1\) changes \( -x^2 \geq -11 \) to \( x^2 \leq 11 \). This step is crucial and something many students tend to forget.
By following these basic principles effectively, the inequality-solving process becomes manageable.
square roots
Taking square roots is an important step in solving quadratic inequalities. Let's look closer.
If you have an equation like \( x^2 \leq 11 \), finding the square root of both sides is necessary to solve for x.
When taking square roots, remember to consider both the positive and negative roots.
For our inequality, \(\begin{align*}&\text{we get} & - \sqrt{11} \leq x \leq \sqrt{11 \),} \end{align*}
This means x can be any value between \( - \sqrt{11 \) and \( \sqrt{11 \).
Always include both roots unless the context specifies only one.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying expressions to solve equations or inequalities.
In our example, starting with \( 11 - x^2 \geq 0 \), we rearranged terms to isolate \( x^2 \).

There's another important aspect: not forgetting how to handle inequalities when dividing or multiplying by negative numbers.
In our example, dividing by -1 required us to flip the inequality sign from \(-x^2 \geq -11\) to \(x^2 \leq 11 \). Thus, being careful with algebraic manipulations, keeping track of steps, and double-checking results are key in mastering problems involving inequalities.

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