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Chapter 1: Problem 16

Solve the equations. a. \(|m+1|=5\) b. \(|m+1|=0\) c. \(|m+1|=-1\)

Short Answer

Expert verified
a. \(4, -6\) b. \(-1\) c. No solution

Step by step solution

01

Understand the Absolute Value Equation

Absolute value equations of the form \(|x| = a\) have two possible solutions when \(a > 0\); that is \(x = a\) or \(x = -a\). When \(a = 0\), we have only one solution, \(x = 0\). When \(a < 0\), there is no solution because the absolute value cannot be negative.
02

Solve \(|m+1|=5\)

For \(|m+1|=5\): \(|m+1| = 5\) implies two equations: \(m+1 = 5\) or \(m+1 = -5\). Solve each equation separately. \m+1 = 5\ leads to \(m = 4\), and \m+1 = -5\ leads to \(m = -6\). Thus, the solutions are \(m = 4\) and \(m = -6\).
03

Solve \(|m+1|=0\)

For \(|m+1|=0\): \(|m+1| = 0\) implies one equation: \(m+1 = 0\). Solving for \(m\), we get \(m = -1\). Thus, the solution is \(m = -1\).
04

Solve \(|m+1|=-1\)

For \(|m+1|=-1\): Since the absolute value cannot be negative, there are no solutions for \(|m+1| = -1\).

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on a number line, regardless of direction. For example, the absolute value of both 5 and -5 is 5. This is always a non-negative number. The absolute value is denoted by vertical bars on each side of the variable or number, like this: \(|x|\). In simple terms, it strips any negative sign and returns the positive magnitude of the number.
Equation Solving
In algebra, solving an equation means finding the value of the variable that makes the equation true. For absolute value equations, we consider the nature of absolute values and solve accordingly:

If you have an absolute value equation in the form \(|x| = a\), where \(a > 0\), it has two possible solutions: \(x = a\) and \(x = -a\).
If \(a = 0\), it means \(|x| = 0\), the only solution is \(x = 0\).
If \(a < 0\), there are no solutions because an absolute value cannot be negative.

For example, consider the equation \(|m+1| = 5\). This can be split into two separate equations: \((m + 1) = 5\) and \((m + 1) = -5\). Solving each, we find \(m = 4\) and \(m = -6\).
No Solution Cases
Absolute value equations can sometimes have no solutions. This occurs when the equation sets the absolute value equal to a negative number. For example, an equation like \(|m + 1| = -1\) has no solution because there is no number that can satisfy this equation. By definition, absolute values are always non-negative. Therefore, equating it to a negative number is impossible, leading to no solution.
In summary, always check the right side of the absolute value equation. If the value is negative, you can immediately conclude that there are no solutions.

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Most popular questions from this chapter

Martha is in a boat in the ocean \(48 \mathrm{mi}\) from point \(A\), the closest point along a straight shoreline. She needs to dock the boat at a marina \(x\) miles farther up the coast, and then drive along the coast to point \(B, 96 \mathrm{mi}\) from point \(A\). Her boat travels \(20 \mathrm{mph}\), and she drives \(60 \mathrm{mph}\). If the total trip took \(4 \mathrm{hr}\), determine the distance \(x\) along the shoreline.

Solve the inequality. Write the solution set in interval notation. $$-7<-\frac{1}{3} z+3$$

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