Problem 52 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|12-6 x|+3 \geq 9$$

Step-by-Step Solution

Answer

x eq 1 or x eq -1

Step 1: Isolate the Absolute Value Expression

Start by isolating the absolute value expression. Subtract 3 from both sides of the inequality: o eq 9 - 3Now, we have |12 - 6x eq 6 Thus, the inequality is |12 - 6x eq 6eq 6.

Step 2: Solve the Resulting Inequalities

The definition of absolute value leads us to split this into two separate inequalities: 1) 12 - 6x eq 6 eq -6 + 12 eq 6 eq x eq x eq -1

Key Concepts

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of the direction. This means absolute value is always non-negative. For any real number \(a\), the absolute value is denoted as \(|a|\) and is defined as:

• \(|a| = a\) if \(a \geq 0\)
• \(|a| = -a\) if \(a < 0\)

Understanding absolute value is crucial, especially when dealing with inequalities, as it helps in comparing distances without considering direction.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. The common forms of inequalities include:

• \(a < b\) (less than)
• \(a \leq b\) (less than or equal to)
• \(a > b\) (greater than)
• \(a \geq b\) (greater than or equal to)

When solving inequalities involving absolute values, we often have to solve two separate inequalities to account for both the positive and negative scenarios of the absolute value expression.

Isolate Absolute Value Expression

The first step in solving an absolute value inequality is to isolate the absolute value expression on one side of the equation. For the given inequality \(|12 - 6x| + 3 \geq 9\), we need to start by isolating \(|12 - 6x|\).

Subtract 3 from both sides: \[|12 - 6x| + 3 - 3 \geq 9 - 3\] \[|12 - 6x| \geq 6\] Once the absolute value is isolated, the inequality becomes simpler to work with, allowing us to proceed further.

Split into Separate Inequalities

After isolating the absolute value expression \(|12 - 6x| \geq 6\), we split the inequality into two separate cases, considering both the positive and negative aspects of the absolute value:

• Case 1: \(12 - 6x \geq 6\)
• Case 2: \(12 - 6x \leq -6\)

Solving each case independently:

For Case 1: \[12 - 6x \geq 6\] \[-6x \geq -6\] \[x \leq 1\] For Case 2: \[12 - 6x \leq -6\] \[-6x \leq -18\] \[x \geq 3\] Combining these results gives us the final solution for \(|12 - 6x| + 3 \geq 9\) as: \[ x \geq 3 \text{ or } x \leq 1\].