Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write the solution set of each equation. $$ |4 x-12|+8=0 $$

Step-by-Step Solution

Answer

No solution, as absolute values cannot be negative.

Step 1: Isolate the Absolute Value

First, subtract 8 from both sides of the equation: \( |4x - 12| + 8 = 0 \rightarrow |4x - 12| = -8 \).

Step 2: Analyze Absolute Value

Recall that the absolute value of any expression is always non-negative. Since \(-8\) is a negative number, \(|4x - 12|\) cannot equal \(-8\).

Step 3: Conclusion

Since an absolute value can't be negative, the equation has no solutions.

Key Concepts

Understanding Absolute Value Equations

Absolute value equations are equations where the unknown variable is inside an absolute value expression. The absolute value, represented by vertical bars like \(|x|\), measures the distance of a number from zero on the number line, so it is always non-negative. This means that an expression like \(|x|\) can never equal a negative number.When solving absolute value equations, you aim to find all possible values of the variable that make the equation true. For example, if you have \[|x| = 5\], this means the variable could be either \(x = 5\) or \(x = -5\), because both numbers are 5 units away from zero on the number line. To effectively solve any absolute value equation, take these steps:

• Isolate the absolute value on one side of the equation.
• Consider both the positive and negative cases by removing the absolute value bars and setting up separate equations.
• Solve each resulting equation individually.

Understanding the nature of absolute values helps in deciding the feasibility of equations, especially when it comes to identifying impossible situations.

Concept of 'No Solution' in Equations

The term 'no solution' refers to when an equation has no value for the variable that can satisfy all parts of the equation. This often happens in equations involving absolute values when the isolated absolute value is set to a negative number.In our original equation example, \[|4x - 12| + 8 = 0\], following the logical steps:

• Subtract 8 from both sides to isolate the absolute expression, yielding \[|4x - 12| = -8\].
• Realize that an absolute value cannot equal a negative number.
• Hence, the conclusion is that no solution exists for this equation, because no real number will satisfy the equation when a negative result is required from taking the absolute value.

Recognizing situations leading to 'no solutions' is vital in algebra, as it prevents futile attempts in finding solutions for impossible conditions.

Isolation of Absolute Value

Isolating the absolute value expression is a crucial first step in solving any absolute value equation. It involves rearranging the equation so that the absolute value expression is by itself on one side of the equation.For instance, in the equation \[|4x - 12| + 8 = 0\]:

• The goal is to have \[|4x - 12|\] alone on one side, so we subtract 8 from both sides to get \[|4x - 12| = -8\].
• Once isolated, check whether the equation is feasible or needs further adjustments.
• If the isolated expression equals a negative number (as in this example), it immediately implies no solution, since an absolute value cannot output a negative.

This isolation technique clarifies the steps needed to attempt solving and quickly evaluates the possibility of solutions for absolute value equations.