Problem 23 - Algebra and Trigonometry

Question

\(23-48\) 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 Inequality

The inequality \(|x| \leq 4\) represents the set of numbers whose distance from zero is less than or equal to 4. This can be represented as two simple inequalities: 1. \(x \leq 4\)2. \(x \geq -4\).

Step 2: Solve the Inequalities Separately

We solve each of the inequalities separately: For \(x \leq 4\), the solution is all real numbers less than or equal to 4.For \(x \geq -4\), the solution is all real numbers greater than or equal to -4.

Step 3: Combine the Solutions

The solution to \(|x| \leq 4\) is the intersection of the solutions to the two inequalities: \(-4 \leq x \leq 4\). This means that \(x\) lies between -4 and 4, inclusive of the boundaries.

Step 4: Express in Interval Notation

The interval notation for the solution \(-4 \leq x \leq 4\) is:\([-4, 4]\). This means that the solution includes all numbers from -4 to 4, including -4 and 4 themselves.

Key Concepts

Interval Notation

Interval notation is an efficient way to describe a range of numbers, especially when dealing with inequalities. When we talk about interval notation, we are essentially describing a set of numbers between two bounds. In the case of the inequality \(-4 \leq x \leq 4\), the interval notation is written as \([-4, 4]\).

• The square brackets \([\underline{\phantom{xxx}} ]\) indicate that the endpoints -4 and 4 are included in the set. This is because the inequality is "less than or equal to," which means that x can be equal to the boundary numbers.
• If the endpoints weren’t included (if, for example, it was \(-4 \lt x \lt 4\)), we would use round brackets \((\text{ )}\) instead.

Using interval notation helps in quickly identifying the range of acceptable values for a variable. It's widely used in mathematics because it's succinct and precise.

Inequality Solutions

To solve inequality problems, especially those involving absolute values, we need to first understand the equality represented. For the inequality \(|x| \leq 4\), we're interested in all values of \(x\) whose absolute value does not exceed 4. Think of it as finding the numbers at most 4 units away from zero on a number line.

• We break down the inequality \(|x| \leq 4\) into two separate linear inequalities: \(x \leq 4\) and \(x \geq -4\).
• By solving these inequalities individually, we find the allowable values for \(x\). These solutions are then combined to form a complete set of numbers that satisfy the original inequality.

This technique is vital because it allows us to systematically address each part of the inequality, ensuring no possible solution is overlooked. The solution's intersection will give us the final answer, often showcased elegantly using interval notation.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operations. In the inequality \(|x| \leq 4\), \(x\) is our variable, and the expression is designed to represent a constraint on the values that \(x\) can take.

• Absolute value expressions like \(|x|\) deal with distance; they show how far the number is from zero, regardless of direction. This makes these expressions particularly useful in defining conditions with spatial implications.
• Algebraic manipulation often involves simplifying or rearranging expressions to isolate the variable of interest, as seen when breaking down \(|x| \leq 4\) into \(x \leq 4\) and \(x \geq -4\).

By understanding algebraic expressions, one can interpret and solve inequalities, ensuring that our solutions fit within the parameters set by the problem. Solving for \(x\) in various mathematical settings is a core skill that builds towards more complex algebraic reasoning.