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Chapter 1: Problem 19

Solve the equations. \(-3=-|c-7|+1\)

Short Answer

Expert verified
c = 11 or c = 3

Step by step solution

01

Isolate the Absolute Value Term

Start by moving the constant term on the right side of the equation to the left side. \( -3 = -|c-7| + 1 \)Subtract 1 from both sides:\( -3 - 1 = -|c-7| \)Simplify:\( -4 = -|c-7| \)
02

Remove the Negative Sign

Divide both sides of the equation by -1 to get rid of the negative sign:\( -4 / -1 = -|c-7| / -1 \)Simplify:\( 4 = |c-7| \)
03

Solve the Absolute Value Equation

Set up two equations to solve for the value of c, as the absolute value expression \( |c-7| \) can be equal to 4 or -4. First equation:\( c-7 = 4 \)Second equation:\( c-7 = -4 \)
04

Solve for c

Solve each equation for c.For \( c-7 = 4 \):Add 7 to both sides:\( c = 4 + 7 \) Simplify:\( c = 11 \)For \( c-7 = -4 \):Add 7 to both sides:\( c = -4 + 7 \)Simplify:\( c = 3 \)

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Key Concepts

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

absolute value equations
Absolute value equations involve the absolute value operation, denoted as \(|x|\). It measures the distance of a number from zero on a number line, always yielding a non-negative result.
For instance, \(|4| = 4\) and \(|-4| = 4\).
When solving absolute value equations, it's essential to recognize that an absolute value equation \( |A| = B \) translates to two possible cases:
  • \( A = B \)
  • \( A = -B \)
This is because the expression inside the absolute value (\( A \)) could originally be positive or negative, but still result in \( B \) when absolute value is taken.
isolation of absolute value terms
To solve absolute value equations, the first step is to isolate the absolute value term on one side of the equation. In the given problem, \(-3 = -|c-7| + 1\), we start by moving the constant term (+1) from the right-hand side to the left. Subtracting 1 from both sides, we get \(-3 - 1 = -|c-7|\), which simplifies to \(-4 = -|c-7|\).
Next, we handle the negative sign. By dividing both sides by -1, we get \(4 = |c-7|\), effectively isolating the absolute value term. It's crucial to isolate the absolute value to clearly see the relationship between the variable and its absolute value.
solving step-by-step
Once the absolute value term is isolated, we can proceed to solve the equation step-by-step by addressing the two cases mentioned earlier. In the equation \(4 = |c-7|\), it translates to:
  • First case: \(c-7 = 4\)
  • Second case: \(c-7 = -4\)
For the first case, solve for \(c\):
Add 7 to both sides: \(c = 4 + 7\).
Simplify: \(c = 11\).
For the second case, solve for \(c\):
Add 7 to both sides: \(c = -4 + 7\).
Simplify: \(c = 3\).
Therefore, the solutions to the absolute value equation are \(c = 11\) and \(c = 3\). This step-by-step approach ensures that all possible values of the variable that satisfy the original equation are found.

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