Question

Solve the inequality. Then graph the solution set on the real number line. \(|x|>8\)

Short answer

The solution set to the given inequality \(|x|>8\) are all numbers less than -8 or greater than 8. In a graph, this can be represented by two line segments on the real number line: one going to the left from -8 (not including -8), the other going to the right from 8 (not including 8).

Step-by-step solution

Solving the Inequality

The given inequality is \(|x|>8\). Absolute value inequalities of the form \(|x|>a\) are equivalent to \(xa\). So, the solution to our inequality will be \(x<-8\) or \(x>8\). This means our solution set is all real numbers less than -8 and greater than 8.

Graphing the Solution

To graph this solution set on the number line, a notation can be used where open dots represent values that are not included in the solution (numbers are not less than or equal to -8 nor greater than or equal to 8, just less than -8 and greater than 8). Fill the number line to the left of -8 and to the right of 8, leaving gaps at -8 and 8 themselves since they are not included in the solution set.

Key concepts

Solving Inequalities

When it comes to solving inequalities, the process can differ slightly from solving standard equations. Inequalities are statements about the relative size or order of two objects, typically involving expressions that are less than, greater than, or not equal to each other. For absolute value inequalities like \(|x| > 8\), the absolute value symbol indicates the distance a number is from zero on the number line, regardless of direction.

To solve \(|x| > 8\), we interpret it as 'the distance of x from 0 is greater than 8'. This leads us to two scenarios: either x is greater than 8 or less than -8. Thus, the solution to the inequality are all values of x that satisfy either condition, meaning all the real numbers that lie to the left of -8 and to the right of 8 on the number line.

It's also important to recognize that with inequalities, flipping the inequality symbol when multiplying or dividing by a negative number is a crucial step, although this particular detail is not necessary for the current problem. Being cautious with these rules ensures the solutions to inequalities are accurate.

Graphing on Number Line

Once the solution to an inequality has been determined, graphing on a number line helps visualize the set of all possible answers. Graphing is particularly useable in displaying the solutions to absolute value inequalities. Starting with a horizontal line representing all real numbers, we then mark the significant boundary numbers identified from solving the inequality.

In the example of \(|x| > 8\), open dots are placed at -8 and 8 to indicate that these exact values are not part of the solution. These open dots are crucial symbols differentiating between 'greater than or equal to' and 'greater than'. The regions less than -8 and greater than 8 are then shaded or filled in to represent all the numbers that are included in the solution set. Graphing these solutions provides a clear and immediate visualization of which numbers are solutions to the given inequality.

Real Numbers

The term real numbers encompasses all possible values that can be represented on a number line, including both rational and irrational numbers. Rational numbers are those that can be expressed as a fraction of integers, while irrational numbers cannot be precisely represented as a simple fraction.

Understanding real numbers is fundamental in mathematics, especially when working with inequalities. The solution to an inequality often consists of a set of real numbers. For instance, the solution to \(|x| > 8\) includes all real numbers that are greater than 8 or less than -8. These numbers have a tangible representation on a number line, which students can readily grasp. This understanding of real numbers is crucial for interpreting the solutions to inequalities and for using real-life applications where such math concepts are applied.