Question

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

Short answer

The solution to the inequality \(x^2 > 4\) is \(x < -2\) or \(x > 2\), graphed as an open dot at \(x = -2\) and \(x = 2\) with lines extending to the left and right respectively.

Step-by-step solution

Solve the Corresponding Equality

In order to find the solution to \(x^2 > 4\), first start by solving the quadratic equation \(x^2=4\). The solutions to this equation will split the number line into intervals which can be tested in the inequality. The equation is solved by taking square root on both sides to get \(x=\pm \sqrt{4}\). This means \(x=2\) and \(x=-2\).

Test the intervals

The solutions \(x=2\) and \(x=-2\) divide the number line into three parts: \(x < -2\), \(-2 < x < 2\), and \(x > 2\). Pick one number from each interval, for instance -3, 0, and 3, and substitute each into the original inequality \(x^2 > 4\). If the inequality holds true for that number, then the entire interval is part of the solution set.

Solution and Graph

After testing, it is determined that the inequality \(x^2 > 4\) holds true for \(x < -2\) and \(x > 2\). Therefore, the solution set is \((-∞, -2) ∪ (2, ∞)\). Since the problem does not indicate or equal to in the inequality, the values -2 and 2 are not included in the solution. Hence open circles must be put on -2 and 2 in the graph on the number line.

Key concepts

Quadratic Equations

Quadratic equations are a cornerstone in algebra. These equations are typically written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. For this particular exercise, we have a quadratic expression \(x^2\) with a comparison sign rather than an equality to zero, specifically \(x^2 > 4\). This forms an inequality rather than a standard quadratic equation.

Solving a quadratic inequality often involves finding the roots of its corresponding quadratic equation, \(x^2 = 4\). These roots, \(x = \pm \sqrt{4}\), simplify to \(x = 2\) and \(x = -2\). Recognizing these specific points is essential because they are where the inequality either switches from true to false, or vice versa. These values help us find intervals on the number line which need further exploration.

Solution Sets

Understanding solution sets is vital when dealing with inequalities. For the inequality \(x^2 > 4\), the goal is to determine where this statement is true based on the quadratic roots we've calculated.

The solutions to the equality \(x^2 = 4\) divide the number line into different intervals: \(x < -2\), \(-2 < x < 2\), and \(x > 2\). By testing these intervals in the inequality \(x^2 > 4\), and noticing which makes the statement true, we can determine the solution set. In our case, the solution set is \((-∞, -2) \cup (2, ∞)\). This means the inequality holds true for values in these two intervals.

Remember, the boundaries (-2 and 2) are not included in the solution set because the inequality is strictly greater than 4, not greater than or equal to.

Number Line Graphing

Graphing solution sets on a number line is a visualization technique that helps illustrate where the variable satisfies given conditions. In this exercise, we graph the solution set \((-∞, -2) \cup (2, ∞)\) on a number line.

This involves marking these intervals, where \(x^2 > 4\) holds true. We use open circles on -2 and 2, since these values are not included in the solution set. The symbol \((- ∞, -2)\) indicates all numbers from negative infinity up to but not including -2, and \((2, ∞)\) indicates all numbers greater than 2 going up to infinity.

• Draw a number line and mark the points -2 and 2.
• Use open circles at -2 and 2 to show they aren't included.
• Shade the area to the left of -2 and the area to the right of 2.

Visualizing the solution set in this way can make the abstract solution more tangible.