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Chapter 2: Problem 3

Solve the equation. \(|4-x|=7\)

Short Answer

Expert verified
The solutions are \(x = -3\) and \(x = 11\).

Step by step solution

01

Understand the Definition of Absolute Value

The absolute value \(|a|\) represents the distance of 'a' from 0 on the number line and is always positive. Therefore, \(|4-x|=7\) means that the expression \(4-x\) can be either 7 or -7.
02

Solve the Positive Case

Assume \(4-x=7\). Solve for \(x\) by subtracting 4 from both sides:\[4-x=7 \-x = 7 - 4 \-x = 3 \x = -3\]
03

Solve the Negative Case

Assume \(4-x=-7\). Solve for \(x\) by subtracting 4 from both sides:\[4-x=-7 \-x = -7 - 4 \-x = -11 \x = 11\]
04

Verify the Solutions

Substitute \(x = -3\) back into the original equation: \[|4 - (-3)| = |4 + 3| = 7\], which holds true.Substitute \(x = 11\) back into the original equation: \[|4 - 11| = |-7| = 7\], which also holds true. Both values satisfy the equation.

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Key Concepts

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

Absolute Value
Absolute value is a fundamental concept in mathematics often symbolized by vertical bars, like this:
  • \(|a|\) represents the absolute value of \(a\).
  • The absolute value denotes the distance from zero on the number line without considering direction.
  • It is always a non-negative value, so \(|a|\) is the positive 'magnitude' of \(a\).
For example,
  • \(|5| = 5\)
  • \(|-5| = 5\)
In the equation \(|4-x|=7\), we understand that \(4-x\) is 7 units away from zero. Therefore, \(4-x\) could either be 7 or -7.
Distance on a Number Line
Distance on a number line is a straightforward way to understand absolute value equations.
  • It focuses on how far an expression is from zero.
  • This makes solving absolute value equations intuitive, as we think about potential 'distances' the number could be.
When we have an equation like \(|4-x| = 7\), imagine asking "How can an expression like \(4-x\) be located 7 units away from zero on a number line?" By visualizing it, you can derive:
  • It can be +7 (right direction) or -7 (left direction).
These distances point to two possible cases for the equation.
Positive and Negative Cases
Solving absolute value equations typically requires exploring both positive and negative scenarios.
  • We start by assuming the expression inside the absolute value can be positive.
  • We then assume that the same expression could also be negative.
In the exercise
  1. First, assume \(4-x=7\):
    By solving, we get \(x = -3\).
  2. Then, consider \(4-x=-7\):
    Solving results in \(x = 11\).
Recognizing both conditions gives all potential solutions for the equation.
Equation Verification
Verification is the final step in solving equations, ensuring that our solutions are accurate.
  • It's critical as it checks if substitutions fulfill the original equation.
  • This prevents errors, confirming the proposed solutions.
Let's verify both solutions of our exercise:
  • Plugging \(x = -3\):
    \(|4 - (-3)| = |4 + 3| = 7\)
  • Substituting \(x = 11\):
    \(|4 - 11| = |-7| = 7\)
Both checks confirm that \(x = -3\) and \(x = 11\) are indeed correct solutions, satisfying the equation.

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Most popular questions from this chapter

The surface area \(S\) of a cube with edge length \(x\) is given by \(S(x)=6 x^{2}\) for \(x>0 .\) Suppose the cubes your company manufactures are supposed to have a surface area of exactly 42 square centimeters, but the machines you own are old and cannot always make a cube with the precise surface area desired. Write an inequality using absolute value that says the surface area of a given cube is no more than 3 square centimeters away (high or low) from the target of 42 square centimeters. Solve the inequality and write your answer using interval notation.

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