Problem 59 - College Algebra

Question

Solve absolute value inequality. \(|x|<3\)

Step-by-Step Solution

Answer

The solution to the inequality \(|x|<3\) is \(-3 < x < 3\).

Step 1 : Dismantle the Absolute Value Inequality

The first thing to do is to break the absolute value inequality \(|x|<3\) into two separate inequalities: \(x < 3\) and \(-x < 3\)

Step 2 : Solve the First Inequality

Solving the inequality \(x < 3\) is straightforward because it is already in a form that can be resolved directly. This gives us the solution \(x < 3\)

Step 3 : Solve the Second Inequality

The second inequality \(-x < 3\) needs to be simplified. By multiplying both sides of the inequality by \(-1\), we change the direction of the inequality and get the solution \(x > -3\)

Step 4 : Combine the Solutions

Having solved both inequalities, the final solution is the combination of both. Given that both inequalities must be true simultaneously (because the absolute value is less than 3 either way), the solution is found where both solutions intersect, which gives us the final result: \(-3 < x < 3\)

Key Concepts

Inequality Solving

Solving inequalities is a crucial skill in algebra. It's a bit like solving equations, but with some additional rules. When you solve an inequality, you're looking for a range of values rather than a single number. The result tells us where the expression holds true.

In this process, the fundamental steps include:

• Isolating the variable to one side.
• Simplifying the expression as much as possible.
• Identifying the solution set—the range of possible solutions for the variable.

Unlike equations, inequalities can have infinite solutions, making them slightly different in nature but just as important. When solving, you must be cautious of the rules, especially when multiplying or dividing by negative numbers, as this changes the inequality direction.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means both positive and negative numbers are treated as their positive counterparts.

The notation for absolute value is two vertical bars, like this: \(|x|\). For example, \(|-5|\) is 5 because the distance from -5 to 0 is 5.

The absolute value notation affects inequalities by creating a "distance boundary." When you encounter an absolute value inequality like \(|x| < 3\), it tells you that the distance of x from zero should be less than 3.

• This creates two scenarios: \(x < 3\) and \(-x < 3\).
• These scenarios must be solved separately and combined to find where both conditions are true.

Understanding absolute values is essential as they find their application in real-world problems where magnitude is more important than direction.

Compound Inequalities

Compound inequalities involve two or more inequalities joined together. They are used to express a range of possible values that a variable can assume.

With absolute value inequalities such as \(|x| < 3\), the solution turns into a compound inequality after breaking down the inequality into two parts: \(x < 3\) and \(-x < 3\).

To solve a compound inequality, you:

• Solve each inequality separately.
• Combine the results.
• Identify the overlap or intersection of the two solutions.

The solution for our example, \(-3 < x < 3\), represents all numbers that are greater than -3 and less than 3. This range contains all possible values x can take. Being comfortable with compound inequalities enables you to tackle complex equations efficiently.