Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{2}-4 x-1>0\)

Short answer

The solution of \(x^{2}-4 x-1>0\) is \(x \in (-\infty , 2- \sqrt{5}) \cup (2+ \sqrt{5}, \infty)\).

Step-by-step solution

Rearrange the Inequality

First, the inequality is already in a standard quadratic form as \(ax^2+bx+c>0\), where \(a=1\), \(b=-4\), and \(c=-1\). So, no further rearrangement is needed.

Find the Critical Points

The critical points are the roots of the equation derived from the inequality. Solve the equation \(x^{2}-4x-1=0\). To find the roots, you can resort to the quadratic formula: \(x= \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\), it results in \(x=\frac{4±\sqrt{16+4}}{2}=\frac{4±\sqrt{20}}{2} = 2± \sqrt{5}\), which are the two critical points.

Determine the Sign on Intervals

Using the two critical points \(2- \sqrt{5}\) and \(2+ \sqrt{5}\), three intervals on the real number line are defined: \(-\infty, 2- \sqrt{5}\), \(2- \sqrt{5}, 2+ \sqrt{5}\), and \(2+ \sqrt{5}, \infty\). Test a number in each interval in the inequality \(x^{2}-4x-1>0\). If a number satisfies the inequality, then the entire interval satisfies the inequality. For example, if we take -10, 0 and 4 which lays respectively in the above intervals and put these in the inequality, it will be found that for the first and third intervals the inequality is true, while for the second it's false.

Graph the Solution Set

On a number line, plot the two points \(2- \sqrt{5}\) and \(2+ \sqrt{5}\) and draw a dashed line considering that the inequality doesn't include the equal sign, hence the end points are not included in the solution set. The solution set is all x-values contained within the intervals \(-\infty, 2- \sqrt{5}\) and \(2+ \sqrt{5}, \infty\). Draw arrow from these points towards -infinity and +infinity to represent these intervals.

Key concepts

Quadratic Inequalities

When dealing with quadratic inequalities such as \(x^2 - 4x - 1 > 0\), we follow a process similar to solving quadratic equations, but with additional steps to find the range of values satisfying the inequality. First, ensure the inequality is structured as \(ax^2 + bx + c > 0\). Once in standard form, the focus shifts to solving the associated quadratic equation \(x^2 - 4x - 1 = 0\) to identify the critical points, or roots. These roots divide the number line into intervals, helping us determine where the inequality holds true. Utilizing a "number test" in these intervals guides us toward solving the inequality, allowing us to determine which intervals make the inequality true, thus identifying the solution set. This process requires understanding the nature of the parabola formed by the quadratic equation, as this shape affects which sections are above or below the x-axis, aligning with our inequality conditions. Keep in mind, critical points derived from the roots are not included in the solution set of a strict inequality.

Real Number Line

The real number line is an essential concept when solving inequalities, particularly quadratic inequalities. It provides a visual means to represent the intervals within which solutions exist. By plotting the critical points obtained from solving the quadratic equation, you can divide the number line into segments. For this exercise, the roots \(2 - \sqrt{5}\) and \(2 + \sqrt{5}\) create three distinct intervals:

• \((-\infty, 2 - \sqrt{5})\)
• \((2 - \sqrt{5}, 2 + \sqrt{5})\)
• \((2 + \sqrt{5}, \infty)\)

Testing a single value from each interval in the original inequality, as demonstrated, verifies whether that entire interval forms part of the solution. On the number line, solutions are represented by continuous lines or arrows that do not include the critical points when dealing with inequalities that don't include equality. Employing symbols like open circles or dashed lines indicates that endpoints are not part of the solution set.

Critical Points

Critical points are crucial in solving quadratic inequalities because they determine where the quadratic expression equals zero. For the inequality \(x^2 - 4x - 1 > 0\), find these points by resolving the equation \(x^2 - 4x - 1 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

• Substitute: \(a = 1\), \(b = -4\), and \(c = -1\)
• Calculate: \(x = \frac{4 \pm \sqrt{16 + 4}}{2}\)
• Simplify: \(x = 2 \pm \sqrt{5}\)

The roots \(2 - \sqrt{5}\) and \(2 + \sqrt{5}\) act as markers that separate the real number line into intervals. These intervals are key to evaluating which regions satisfy the inequality. Testing points from each interval helps determine the solution set, highlighting the importance of understanding and correctly identifying critical points.