Problem 56 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|12-9 x| \geq-12$$

Step-by-Step Solution

Answer

The solution is all real numbers.

Step 1: Understand the Absolute Value Property

Recall that absolute values are always non-negative, meaning \(|a| \geq 0\). Therefore, any absolute value, regardless of the expression inside, will always be greater than or equal to zero. In this case, \(|12-9x| \geq -12\) is always true because absolute values are never negative.

Step 2: No Further Simplification Needed

Because the inequality \(|12-9x| \geq -12\) is always true, there is no need for further simplification. Any value of \(x\) will satisfy the inequality.

Key Concepts

Absolute Value Properties

The absolute value of a number is its distance from zero on the number line. This means it is always a non-negative value. For any real number, \(a\), the absolute value is denoted as \(|a|\).
Here are some important aspects of absolute value properties:

• \(|a| \geq 0\): Absolute value is always non-negative.
• \(|a| = a\) if \(a \geq 0\); and \(|a| = -a\) if \(a < 0\).
• Absolute values can simplify the comparison of distances between numbers.

In the given exercise, the expression \(|12-9x|\) represents the absolute value, which is always non-negative. Thus, it remains greater than or equal to zero.

Inequalities

Inequalities are mathematical expressions involving the symbols \(>\text{,}<\text{,}\text{>=}\text{,}\text{<=}\).
They compare two values or expressions and show the relationship between them. When dealing with inequalities that involve absolute values, it is essential to consider the properties of absolute values.
In the given problem, \(|12-9x| \geq -12\), we observe an inequality involving an absolute value. Given that absolute values are always non-negative, any absolute value will automatically satisfy the condition to be greater than or equal to any negative value, like \(-12\) in this context. This simply means any value replacing \(x\) will satisfy the inequality since the initial statement is always true.

Non-negative Values

The concept of non-negative values is crucial in understanding absolute values and certain types of inequalities. A non-negative value is any number that is greater than or equal to zero.
This includes positive numbers and zero but never negative numbers.

Key points to remember:

• Non-negative values: \(x \geq 0\)
• The range of non-negative values contains zero and all positive numbers.
• Absolute values are a classic example where results are always non-negative.

Applying this to our problem, the fact that \(|12-9x|\) is non-negative means it can never be less than zero, thus it's naturally always greater than or equal to \(-12\). Thereby, making this inequality true for any given value of \(x\).