Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{2}<5\)

Short answer

The solution to the inequality \(x^{2}<5\) is \(-\sqrt{5} < x < \sqrt{5}\). This means that any x-value within this interval will satisfy the inequality.

Step-by-step solution

Solve the related equation

The inequality given is \(x^{2}<5\). To solve this inequality, first, the related equation is solved i.e., \(x^{2}=5\). The square root of 5 should be taken into account in both positive and negative values since we are dealing with a squared variable. Thus, the solutions are \(x= -\sqrt{5}\) and \(x=\sqrt{5}\).

Determine the intervals

Having the solutions as \(x= -\sqrt{5}\) and \(x= \sqrt{5}\), the numbers divide the number line into three intervals: \(-\infty < x < -\sqrt{5}\), \(-\sqrt{5} < x < \sqrt{5}\), and \(\sqrt{5} < x < \infty\). Next, select test points in each interval and substitute into the original inequality.

Test the intervals

Choose a test point from each of the intervals. For example, select -6 for the first interval, 0 for the second interval, and 6 for the third interval, and then substitute these into the inequality \(x^{2}<5\), the inequalities become: \((-6)^2 < 5\) is false, \(0^2 < 5\) is true, and \(6^2 < 5\) is false.

Graph the solution

The only true inequality from the test points is from the interval \(-\sqrt{5} < x < \sqrt{5}\). Therefore, this is the solution interval. The solution interval can be graphed on a number line. Open dots are used at \(-\sqrt{5}\) and \(\sqrt{5}\) because the original inequality does not include the endpoints.

Key concepts

Quadratic Inequalities

Quadratic inequalities involve expressions in the form of ax^2 + bx + c < 0, > 0, \( \leq \) 0, or \( \geq \) 0, where a, b, and c are real numbers and a is not equal to zero. Unlike simple equations, inequalities do not have just specific solutions, but rather a range of solutions that satisfy the condition. To solve these, one must find the values of x that make the inequality true.

The process often involves turning the inequality into a related equation (as seen in the exercise with x^2 = 5) and then understanding how the inequality behaves in different segments of the number line. This method helps visualize which sections of the number line contain the solutions to the inequality.

Number Line Graphing

Graphing on a number line is a way to visually represent the solution set of an inequality. For quadratic inequalities, once the critical points (such as \( -\sqrt{5} \) and \( \sqrt{5} \) from the related equation) are found, these points are marked on the number line, serving as boundaries that divide the line into intervals.

In the given exercise, open circles are placed at \( -\sqrt{5} \) and \( \sqrt{5} \) to indicate that these points are not included in the solution set (since the inequality is '<' and not '\( \leq \)'). The intervals between and outside these points represent potential solution ranges that need to be tested to find where the inequality holds true.

Test Points Method

The test points method is a systematic way to determine which intervals on the number line satisfy the inequality. After identifying the intervals using the solutions from the related equation, you must choose a 'test point' from each interval to substitute back into the original inequality.

For instance, in our given problem, we choose -6, 0, and 6 as test points for the three intervals. When these are substituted into the inequality x^2 < 5, we find which intervals make the inequality true. If the test point satisfies the inequality, the entire interval from which the test point was chosen is included in the solution set. This method confirms that the values between \( -\sqrt{5} \) and \( \sqrt{5} \) satisfy the original inequality.

Square Root Solutions

Solving quadratic inequalities may involve finding square root solutions when the inequality is simplified to an expression where the variables are squared, such as x^2 < k or x^2 > k. To solve for x, you need to take the square root of k, remembering to include both the positive and negative roots since squaring either would give the original k value.

In our exercise, solving the equation x^2 = 5 leads to \( x = \pm\sqrt{5} \), providing us with the critical values for our number line. It's critical to account for both the positive and negative square root solutions as these values delineate the boundary of the solution set on the number line.