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Chapter 1: Problem 62

Solve each absolute value equation or indicate the equation has no solution. $$ |x+1|=5 $$

Short Answer

Expert verified
The solutions to the absolute value equation |x + 1| = 5 are \(x = 4\) and \(x = -6\).

Step by step solution

01

Solve for the positive solution

We first consider the case of (x + 1) being 5 units to the right of zero. This means setting the expression inside the absolute value, (x + 1), equal to 5, then solving for \(x\). So, \(x + 1 = 5\) leads to \(x = 5 - 1 = 4\).
02

Solve for the negative solution

Next, we consider the case of (x + 1) being 5 units to the left of zero. This means setting the expression inside the absolute value, (x + 1), equal to -5, then solving for \(x\). So, \(x + 1 = -5\) leads to \(x = -5 - 1 = -6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Absolute Value
The absolute value of a number refers to its distance from zero on the number line, regardless of direction. Absolute value equations are special kinds of equations in which an expression within absolute value bars must equal a non-negative number. For instance, consider the equation \(|x+1|=5\). To find solutions, you'll work with two possible scenarios: one where the expression inside the absolute value is equal to a positive 5, and another where it is equal to a negative 5. Both cases are necessary because absolute value represents distance without considering direction. It's useful to start by setting up and handling these scenarios individually to solve the equation.
Positive Solution
Solving for a positive solution involves considering the situation where the expression inside the absolute value equates to a positive number. In the given equation \(|x+1|=5\), we first assume that \(x+1\) is equal to 5. This scenario is straightforward:
  • Set \(x+1=5\)
  • Subtract 1 from both sides to solve for \(x\), so \(x=4\)
Thus, \(x=4\) is one of the solutions to the equation. This solution recognizes that the expression inside the absolute value is *exactly* 5 units to the right of zero on the number line.
Negative Solution
Now, let's consider the negative solution, which addresses the scenario where the expression inside the absolute value must be equal to a negative number. For the equation \(|x+1|=5\), we expect \(x+1\) to also be -5. Though it might sound counterintuitive initially, it acknowledges the direction of travel on the number line:
  • Set \(x+1=-5\)
  • Subtract 1 from both sides to solve for \(x\), yielding \(x=-6\)
Thus, \(x=-6\) is another solution to the equation. This solution indicates the expression inside the absolute value is 5 units to the left of zero. So, any absolute value equation matches both a positive and a negative solution.
No Solution in Equations
In some cases, an absolute value equation might not have any solutions. This typically occurs when the absolute value is set to equal a negative number, which is impossible since absolute value represents distance and cannot be negative.Suppose we had an equation like \(|x+1|=-3\). No real number \(x+1\) could satisfy this because distances cannot be negative. Thus, such equations are described as having no solution. In other words, if your absolute value equation asks for an output less than zero, it's a hint that you won't find any solutions whatsoever. This is a helpful check when verifying solutions: ensuring all distances asked by absolute values are realistically possible.

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Most popular questions from this chapter

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{3}{4+i}$$

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