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Chapter 2: Problem 14

Solve the inequality. Then graph the solution set. $$x^{2} \leq 16$$

Short Answer

Expert verified
The solution to the inequality \(x^{2} \leq 16\) is \(-4 \leq x \leq 4\) and the resultant graph is a shaded region on the x-axis, between and inclusive of -4 and 4.

Step by step solution

01

- Identify the inequality

The given inequality is \(x^{2} \leq 16\)
02

- Solve for x

To find the value of x, take the square root of both sides of the inequality. So, \(x\) is less than or equal to \(+\sqrt{16}\) and greater than or equal to \(-\sqrt{16}\), which gives us: \(-4 \leq x \leq 4\)
03

- Identify the solution set

The solution set includes all values of x that satisfy the inequality \(-4 \leq x \leq 4\)
04

- Graph the solution

The graph should be a closed circle at \(x = -4\) and \(x = 4\) and all points between these two on the x-axis should be shaded to indicate all allowable x values between -4 and 4 inclusive.

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Key Concepts

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

Solution Set
In the context of quadratic inequalities, the solution set refers to all the values of the variable that satisfy the given inequality. For instance, when working with the inequality \(x^2 \leq 16\), the solution set consists of all numbers \(x\) that do not cause the inequality to become false. This means we're looking for all \(x\) where \(x^2\) is equal to or less than 16.
To find this, we solve for \(x\) by taking the square root of 16. The solution is where \(x\) ranges between the negative and positive square roots: \(-4 \leq x \leq 4\).
Thus, every number including -4 up to and including 4 is part of the solution set.
  • A solution set can sometimes be expressed in interval notation: \([-4, 4]\), which captures all inclusive values within the bounds.
  • Remember, each value inside the solution set will satisfy the original inequality when substituted back into \(x^2 \leq 16\).
Solving Inequalities
When solving quadratic inequalities such as \(x^2 \leq 16\), your primary goal is to isolate the variable \(x\) on one side of the inequality. Here's a simple method to get there:
1. Recognize the structure: The equation \(x^2 = 16\) corresponds to a scenario where equality holds.
2. Solve for critical points: From \(x^2 = 16\), you find \(x = \pm 4\) by taking the square roots.
3. Evaluate inequality range: Since the inequality involves \(x^2 \leq 16\), reflect on what this means numerically. The values for \(x\) range from \(-4\) to \(4\), inclusive, because \(x^2\) is smallest at zero and grows symmetrically around it.
  • By isolating \(x\), a good practice is to consider what happens when you test values within this range in the inequality. Doing this helps confirm the solution set.
  • Being cautious with the math operations is crucial: squaring and square roots affect the sign, a common mistake for many learners.
Graphing Inequalities
Graphing a quadratic inequality like \(x^2 \leq 16\) allows for a visual representation of the solution set on a number line. To graph:
1. Start by marking the critical points found from solving the inequality, in this case, \(x = -4\) and \(x = 4\). These points are where \(x^2 = 16\).
2. Since the inequality is "less than or equal to", use closed circles at these points. A closed circle means that the point itself is included in the solution set.
3. Shade the region between these points on the number line. This shaded area represents all the values that \(x\) can take, fulfilling \(-4 \leq x \leq 4\).
  • Graphing not only illustrates the permissible range but helps confirm if your calculated solution set matches visually what the inequality describes.
  • Always remember that the direction of shading (left or right from the center of \(x\)) varies based on whether you're dealing with \(\leq\) or \(<\), and identifying critical points is key.

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Most popular questions from this chapter

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Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). Then check your results algebraically by writing the quadratic function in standard form. $$f(x)=-\left(x^{2}+2 x-3\right)$$

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