Problem 73 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x+1|+5=3$$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Isolate the absolute value expression

The first step is to isolate the absolute value expression on one side of the equation. Since we have a \(+5\) on the side of the absolute value term, we subtract 5 from both sides of our equation to get: \(|x+1|=3-5=-2\).

Step 2: Assess the situation

Now we have an absolute value equal to a negative number. The absolute value of a number is always nonnegative, meaning it is always greater than or equal to 0. Therefore, there is no real number that we could substitute for \(x\) that would make our equation true.

Key Concepts

Solving Absolute Value Equations

When faced with an absolute value equation, the goal is to find the values of the variable that make the equation true. An absolute value equation is an equation that contains an absolute value expression. Absolute value refers to the "distance" a number is from zero on a number line, regardless of direction. Hence, it's always positive or zero.

To solve absolute value equations, follow these steps:

• Isolate the absolute value expression.
• Once isolated, set up two separate equations: one for the positive version of the expression, and another for the negative.
• Solve each of these equations separately.

This approach helps us account for both possibilities contributing to the absolute value. Thus, we ensure we are considering all potential solutions.

Isolation of Absolute Value

The first step in solving absolute value equations is isolation. Isolation ensures that the absolute value expression stands alone on one side of the equation. This means you need to manipulate the equation so that anything else is moved to the opposite side.

For instance, in the problem \(|x+1|+5=3\), we start by moving the constant term "+5" to the other side of the equation by subtracting 5 from both sides. This isolates the absolute value, leading to \(|x+1|=3-5=-2\). Once isolated, the equation can be analyzed further.

Absolute Value Properties

Understanding the unique properties of absolute value can help solve equations and recognize "no solution" scenarios.

• The absolute value is always non-negative, meaning it cannot be less than zero. It describes a distance which is a positive quantity or zero.
• Since it's non-negative, any absolute value equation where the expression equals a negative number has no real solution.

In absolute value problems like \(|x+1| = -2\), seeing the negative result immediately signals that no solution exists, due to absolute value properties.

No Solution Cases

Not all equations promise a solution, especially when dealing with absolute values. In some cases, after performing isolation and solving steps, you may encounter an impossibility.

A "no solution" case occurs when, after isolating the absolute value, it equals a negative number. This is inherently impossible because, as discussed, absolute values are non-negative by nature. Therefore, logical contradictions arise which mean there are no real values of the variable that satisfy the equation.

In the example \(|x+1| = -2\), the equation signals no permissible value for \(x\) can satisfy this condition. Understanding and identifying these scenarios is key to mastering absolute value equations.