Problem 27 - Precalculus: Functions and Graphs

Question

Solve the equation. \(3|x+1|-2=-11\)

Step-by-Step Solution

Answer

No solution. The equation is invalid as absolute values cannot be negative.

Step 1: Simplify the Equation

First, let's isolate the absolute value expression. Add 2 to both sides of the equation to get \[3|x+1| = -11 + 2\]which simplifies to \[3|x+1| = -9.\]

Step 2: Divide by the Coefficient of the Absolute Value

To further simplify, divide every term by 3 to isolate the absolute value:\[|x+1| = \frac{-9}{3}\]which simplifies to\[|x+1| = -3.\]

Step 3: Analyze the Absolute Value Condition

The absolute value expression \(|x+1|\) is always non-negative by definition. Therefore, it cannot be equal to -3. Hence, there are no values for \(x\) that satisfy this equation.

Key Concepts

Solving Equations

When tackling any math problem, one of the first things you must understand is the process of solving equations. This involves finding the value of the variable that makes the equation true. Typically, equations are set up to state that two expressions are equal. In this specific problem, we're dealing with an absolute value equation, which adds a layer of complexity.

While working through an equation like this, a structured approach is essential. The three main stages are usually: isolating the expression with the variable, simplifying the equation, and solving for the variable. Each of these stages is crucial for arriving at the correct solution. Remember, the aim is to transform the original equation into a simpler one that is easier to solve.

But equations with absolute values can sometimes lead to unexpected results, as we'll see in this exercise.

Isolating Terms

A vital step when solving equations is isolating terms. This means trying to get the variable or the expression involving the variable by itself on one side of the equation. In our equation, we start by looking at \(3|x+1|-2 = -11\). To isolate the absolute value expression, we add 2 to both sides.

• This gives us \(3|x+1| = -9\).
• By doing this, we simplify the equation and make it easier to work with.

Next, we need the absolute value part, \(|x+1|\), all alone. We achieve this by dividing both sides of the equation by 3.

• So, \(|x+1| = \frac{-9}{3}\), which simplifies to \(|x+1| = -3\).

At this point, it is typically where we'd solve for the variable inside the absolute value, but the result here points out a unique situation.

Non-negative Property of Absolute Value

The concept of absolute value is crucial in understanding why certain equations might not have any solutions. By definition, the absolute value of any expression represents its distance from zero on the number line, making it inherently non-negative. In simpler terms, this means that the outcome of any absolute value calculation can never be negative.

So, in the equation we are working with, we end up with \(|x+1| = -3\). This is a red flag because it contradicts the non-negative property of absolute values. It's impossible for any expression's absolute value to equal a negative number, given that absolute values range from zero to positive numbers only.

This situation leads us to conclude that no real number exists for \(x\) to satisfy this equation. Hence, the equation has no solution, which is an important concept to recognize when dealing with absolute value equations. Understanding these properties ensures you don't waste time trying to solve an equation that fundamentally can't be solved in the real number system.