Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |-x|-|4| \leq 2 $$

Short answer

The solution in interval notation is \[ [-6, 6] \].

Step-by-step solution

Understand the Absolute Value Property

The absolute value of a number, denoted \( |a| \), represents the distance of the number from 0 on the number line, regardless of direction. Therefore, \( |a| = | -a| \) for any real number \( a \).

Simplify the Inequality

Replace \( |4| \) with 4, as the absolute value of 4 is 4. Thus, we have \( |-x| - 4 \leq 2 \).

Simplify the Absolute Value

Since \( |-x| \) is the same as \( |x| \), we can rewrite the inequality as \( |x| - 4 \leq 2 \).

Isolate the Absolute Value Term

Add 4 to both sides of the inequality to isolate the absolute value term: \( |x| \leq 6 \).

Solve for the Variable

By the definition of absolute values, \( |x| \leq 6 \) means \( -6 \leq x \leq 6 \).

Express Solution in Interval Notation

The solution \( -6 \leq x \leq 6 \) can be written in interval notation as \[ [-6, 6] \].

Graph the Solution Set

To graph the solution set \[ [-6, 6] \], draw a number line with a closed circle at -6 and another closed circle at 6, and shade the region in between these two points.

Key concepts

Absolute Value

Absolute value is a fundamental concept in mathematics that measures the distance of a number from zero on the number line. It is always a non-negative number. Absolute value is denoted by vertical bars, like this: \( |x| \). For example, both \( |5| \) and \( |-5| \) equal 5 because both 5 and -5 are five units away from zero.
In our exercise, understanding that the absolute value of \( -x \) is the same as \( |x| \) allows us to simplify inequalities effectively. Once we isolate the absolute value term, we solve the inequality by considering both the positive and negative outcomes.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It uses brackets and parentheses to show the start and end points of an interval.

• Closed intervals, where the endpoints are included, are denoted by square brackets \[ [] \]
• Open intervals, excluding the endpoints, are denoted by parentheses \[ () \]

For example, the interval \[ -6, 6 \] includes all numbers from -6 to 6, inclusive. This is crucial to express solutions in a concise format. In our exercise, we found that \[ -6 \leq x \leq 6 \] translates directly to the interval notation \[ [-6, 6] \].

Graphing Inequalities

Graphing inequalities involves representing their solutions visually on a number line. Here’s a guide on how to graph the solution set \[ [-6, 6] \] from our exercise:

• Draw a horizontal number line.
• Mark the points -6 and 6 on this number line.
• Since the interval is closed (including -6 and 6), use closed circles at these points.
• Shade the region between these points to represent all values \ x \ that satisfy -6 ≤ x ≤ 6.

This visual tool helps to see the range of solutions at a glance. Graphing inequalities is essential for understanding the scope of solutions and how they relate to the number line.