Problem 31 - Algebra 2

Question

Solve each equation. Check your solutions. \(-12|9 x+1|=144\)

Step-by-Step Solution

Answer

The equation has no solution because an absolute value cannot equal a negative number.

Step 1: Isolate the Absolute Value

Start by dividing both sides of the equation by -12 to isolate the absolute value expression. This gives us \[|9x + 1| = -12\]Since an absolute value cannot equal a negative number, this equation has no solution.

Step 2: Verify Your Solution

Since we found in the previous step that the absolute value cannot equal a negative number, we conclude that the equation has no solution. Therefore, there's nothing to check further.

Key Concepts

Absolute Value Equations

Absolute value equations are mathematical expressions where the absolute value of a variable is equated to a number. The absolute value of a number refers to its distance from zero on a number line, always expressed as a non-negative. This means that any expression inside the absolute value signs can have both a positive and a negative solution. For example,

• If we have \( |x| = 5 \), \ then \ either \(x = 5 \) or \(x = -5\).

It's essential to understand that absolute values always yield non-negative results. Whether dealing with whole numbers, fractions, or variables, the same principles apply.
In an equation like \( -12|9x+1|=144 \), after isolating and simplifying the absolute value, it must resolve to a non-negative number.

Solving Equations

Solving equations involves finding a value or values for the variables that make the equation true. For absolute value equations, you'll typically follow these steps:

• Isolate the absolute value expression.
• Set up two separate equations to account for both the positive and negative solutions of the absolute value.

In our example, \[ |9x+1| = -12 \], the task is to solve for \ x \ after isolating the absolute value expression by dividing both sides by -12. However, we immediately realize that the absolute value cannot equal a negative number, indicating the equation may not have a feasible solution.

Checking Solutions

Once we find a potential solution, it's crucial to verify it by plugging it back into the original equation. This step ensures that no calculation errors impact the validity of the solution.
For absolute value equations, substitute both possible solutions back into the equation if the absolute value expression could be either positive or negative. Since any correctly solved absolute value equation will hold for both cases, verification is a useful sanity check.
In the example \[ -12|9x+1|=144 \], there were no calculated values for \ x \ since the equation initially led to a mathematical impossibility—an absolute value equating a negative number.

No Solution Concepts

Understanding the concept of "no solution" in algebra is vital, especially with absolute value equations. There are certain scenarios where an equation will not have any viable solutions. Specifically, with absolute values, this occurs if:

• The isolated absolute value expression is set equal to a negative number.

For instance, the equation \[ |9x+1| = -12 \] exposes a critical mathematical principle: absolute values cannot equate to a negative value.
Therefore, when isolating the absolute value in the equation, if it results in a negative number, it is concluded that there's no solution. This is an important concept in algebra, emphasizing the non-negative nature of absolute values.