In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x + 1| + 5 = 3

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 5 from both sides of the equation: |x + 1| + 5 - 5 = 3 - 5.
2. This simplifies to |x + 1| = -2.
3. Recall that the absolute value of any expression is always non-negative, meaning it cannot be negative.
4. Since |x + 1| = -2 is a negative number, this equation has no solution.
5. Conclude that there are no values of x that satisfy the given equation.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases based on the definition.

Solving Absolute Value Equations

To solve an absolute value equation, you typically isolate the absolute value expression and then set up two separate equations: one for the positive case and one for the negative case. For instance, if |x| = a, then x = a or x = -a. This method allows you to find all possible solutions that satisfy the original equation.

No Solution Cases

In some scenarios, an absolute value equation may have no solution. This occurs when the equation leads to a contradiction, such as a negative value equating to an absolute value. For example, if you isolate the absolute value and find that it equals a negative number, it indicates that there are no real solutions to the equation.