Problem 32 - Algebra 2

Question

Solve each equation. Check your solutions. \(1=|5 x+9|+6\)

Step-by-Step Solution

Answer

No solutions exist because absolute values can't be negative.

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression on one side of the equation. The given equation is \( 1 = |5x + 9| + 6 \). To isolate the absolute value, subtract 6 from both sides of the equation: \\[-5 = |5x + 9|\]

Step 2: Recognize the Condition

An important property of absolute values is that they cannot be negative. Since \(-5\) is a negative number, there are no real solutions to the equation. \[\] If \(|5x + 9| = -5\), this equation has no solutions because the absolute value of any real number is non-negative.

Step 3: Conclusion

As there are no values of \(x\) that can satisfy \(|5x + 9| = -5\), the equation \( 1 = |5x + 9| + 6 \) has no solutions.

Key Concepts

Equation Solving

Solving equations involves finding the value of a variable that makes the equation true. In the exercise given, we are dealing with an absolute value equation. The steps for solving such an equation require isolating the absolute value expression. This means we need to perform operations to get the absolute value on one side of the equation on its own.
In our example, the initial equation is given by \(1 = |5x + 9| + 6\). To isolate \(|5x + 9|\), we subtract 6 from both sides, resulting in \(-5 = |5x + 9|\). By writing the expression this way, we hope to proceed to the next steps of exploring possible solutions. However, in this particular instance, another significant factor comes into play as we solve the equation: the nature of absolute values.

Real Solutions

When finding real solutions to an equation, we aim to identify values for the variable where the equation holds true. Real solutions are numbers that we can visualize on the number line and can be positive numbers, negative numbers, or zero.
In our equation \( -5 = |5x + 9| \), identifying real solutions means finding \(x\) where this equality is true. However, because the equation involves an absolute value, we immediately run into a roadblock. Absolute values always result in non-negative outcomes, implying that an absolute value never equals a negative number. Therefore, in this situation, there are no real solutions due to the nature of absolute values. When you encounter such equations, it's essential to recognize that not all equations have real solutions, especially if absolute values are involved.

Properties of Absolute Value

The properties of absolute value are fundamental to understanding certain types of equations. An absolute value of a number essentially reflects its distance from zero on the number line, disregarding direction. This inherent property means the absolute value is always a non-negative number.
For the equation \(|5x + 9| = -5\), it's crucial to understand that it does not align with what we know about absolute values.
Key properties of absolute values include:

• Absolute values are always greater than or equal to zero: \(|x| \geq 0\)
• If \(|x| = a\), then \(x = a\) or \(x = -a\) when \(a \geq 0\)

Given these properties, our equation \(|5x + 9| = -5\) is unsolvable in the realm of real numbers, because there isn't a real number whose absolute value can be negative. This helps us understand why \(-5 = |5x + 9|\) results in no solutions. Understanding these properties helps prevent common mistakes when working with absolute value equations.