Problem 19 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x| \leq 4 $$

Step-by-Step Solution

Answer

The solution is \([-4, 4]\).

Step 1: Understanding Absolute Value Inequalities

An absolute value inequality of the form \(|x| \leq a\) describes all values of \(x\) that are within a distance of \(a\) units from 0 on the number line. This translates to the compound inequality \(-a \leq x \leq a\).

Step 2: Apply the Concept to the Given Inequality

For the inequality \(|x| \leq 4\), we use the rule from Step 1. We can rewrite this as a compound inequality: \(-4 \leq x \leq 4\).

Step 3: Convert to Interval Notation

The solution to the inequality \(-4 \leq x \leq 4\) can be expressed in interval notation as \([-4, 4]\). This indicates that \(x\) includes all values from -4 to 4, inclusive.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. It is always a non-negative number. When dealing with the absolute value, we focus on how far a number is from zero, rather than whether it is to the left or right on the number line.

In mathematical terms, the absolute value of a number \(x\) is represented as \(|x|\). For example, the absolute value of both \(-3\) and \(3\) is \(3\). This concept is crucial when dealing with inequalities, as it requires us to consider two scenarios:

• When \(x\) is positive or zero
  
• When \(x\) is negative

Considering these scenarios allows us to break down absolute value inequalities into compound inequalities, which we solve to find the range of possible solutions.

Interval Notation

Interval notation is a way of representing a set of numbers that represent solutions to an inequality or a range of values. This concise form uses parentheses and brackets to show which values are included in or excluded from the set.

Here’s how it works:

• Parentheses \(()\) indicate that an endpoint number is not included in the interval. These are used for strict inequalities such as \(<\) or \(>\).
• Brackets \([]\) indicate that an endpoint number is included in the interval. These are used for inclusive inequalities such as \(\leq\) or \(\geq\).

For example, the interval \([-4, 4]\) includes all numbers from \(-4\) to \(4\), and includes both \(-4\) and \(4\). This is how you will express the solution to an inequality like \(-4 \leq x \leq 4\) in a simple, clear way.

Interval notation gives a quick way to convey a wide range of numbers, which is especially helpful in mathematical solutions.

Compound Inequalities

Compound inequalities are statements that involve two separate inequalities joined by the word "and" or "or". They describe a set of solutions that satisfy both inequalities at the same time.

An "and" compound inequality takes the form of \(a \leq x \leq b\), meaning \(x\) must satisfy both \(a \leq x\) and \(x \leq b\) at the same time. This results in a limited window of possible values for \(x\). The interval of solutions fall between the minimum and maximum bounds specified by the inequalities.

Consider the inequality \(-4 \leq x \leq 4\). It is a compound inequality that requires \(x\) to be greater than or equal to \(-4\) and less than or equal to \(4\). Solving these inequalities simultaneously gives us a range of values which can be expressed in interval notation as \([-4, 4]\).

In conclusion, understanding how to work with compound inequalities is vital for manipulating and interpreting inequalities involving absolute values.