Problem 106 - Algebra for College Students

Question

Discussion Explain how to write an absolute value inequality as a compound inequality.

Step-by-Step Solution

Answer

Rewrite \( |x| < a \) as \(-a < x < a \).

Step 1: Understand Absolute Value

Absolute value of a number represents its distance from zero on the number line, regardless of direction. For any number x, the absolute value is denoted as \(|x|\).

Step 2: Formulate Absolute Value Inequality

Consider the absolute value inequality \( |x| < a \), where \( a \) is a positive real number. This describes all values of \( x \) whose distance from zero is less than \( a \).

Step 3: Write as a Compound Inequality

The inequality \( |x| < a \) can be rewritten as two separate inequalities: \( -a < x < a \). This compound inequality shows that \( x \) lies between \( -a \) and \( a \).

Step 4: Example Analysis

For example, if you have \( |x-3| \leq 5 \), rewrite this as \(-5 \leq x - 3 \leq 5 \). Then, solve both inequalities to get \( -2 \leq x \leq 8 \).

Key Concepts

compound inequality

A compound inequality involves two separate inequalities that are combined into one statement by either 'and' or 'or'. When dealing with absolute value inequalities, we use compound inequalities to describe a range of values that satisfy the original condition.

For instance, consider the absolute value inequality \(|x| < 4\). This means that the distance of x from zero is less than 4. To convert this into a compound inequality, we need to split the absolute value expression into two parts:

• \(x < 4\)
• \(x > -4\).

Combining these, we get the compound inequality: \(-4 < x < 4\). This tells us that x can be any value between -4 and 4, but not including -4 and 4 themselves.

If we had \(|x| \leq 4\), the compound inequality would be \(-4 \leq x \leq 4\), meaning x can be any value from -4 to 4, including -4 and 4. Remember, compound inequalities help describe all the possible solutions to the inequality.

number line

A number line is a visual representation of numbers placed in a straight line where every point corresponds to a number. This tool is extremely useful in understanding absolute value inequalities. When we plot points on a number line, we can instantly see how far they are from each other or from zero.

For the earlier example, \(|x| < 4\), we can plot -4 and 4 on the number line and mark all the points between them. This visualizes the range of x that lies between -4 and 4.

Let's consider \(|x-3| \leq 5\). To interpret this on a number line, first recognize that our inequality is centered around 3 (not zero). So, we mark 3 on the number line and plot points 5 units away from it, resulting in -2 and 8. Now, the points between -2 and 8 (including -2 and 8) show all possible values of x. This way, a number line helps in better understanding and solving inequalities.

distance from zero

Absolute value represents the distance from zero, which is always a non-negative number. For any number x, \(|x|\) tells us how far x is from zero without considering direction. Whether x is positive or negative, its absolute value is always positive.

For example, \( |3| = 3 \), and \( |-3| = 3 \), both of which are 3 units away from zero.

In an absolute value inequality like \(|x| < 4\), we are essentially looking for all values of x whose distance from zero is less than 4 units. This concept is central to understanding and solving absolute value inequalities since it shifts our focus from the exact numbers to how far they are from the center point (usually zero).

By converting these inequalities into compound inequalities, we can easily describe the solutions in terms of a range, providing a more intuitive grasp of the values that satisfy the inequality.