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Chapter 1: Problem 36

For what value(s) of \(x\) is \(|x|=4\) true?

Short Answer

Expert verified
The values of \( x \) are \( 4 \) and \( -4 \).

Step by step solution

01

Understand the Absolute Value Function

The absolute value of a number is its distance from zero on the number line, without regard to direction. Therefore, \(|x| = 4\) means that the distance of \(|x|\) from zero is 4.
02

Set Up Two Equations

Given \(|x| = 4\), the number \ (x) \ can be either positive or negative. This gives us two equations: \( x = 4 \) and \( x = -4 \).
03

Solve Each Equation

Solving the first equation: \( x = 4 \). Solving the second equation: \( x = -4 \).
04

Verify the Solutions

Check both solutions by substituting back into the absolute value equation. \( |4| = 4 \) and \( |-4| = 4 \), both are true.

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Key Concepts

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

absolute value function
The absolute value function is a fundamental mathematical concept that measures the distance of a number from zero on a number line. It is always a non-negative number. For any real number \( x \), the absolute value is denoted as \( |x| \).

Here are a few key points:
  • If \( x \) is positive, \( |x| = x \).
  • If \( x \) is negative, \( |x| = -x \).
  • If \( x = 0 \), \( |x| = 0 \).

For example, \( |4| = 4 \) and \( |-4| = 4 \), because both 4 and -4 are 4 units away from zero.
distance from zero
Another way to understand the absolute value function is to think of it as the distance a number is from zero on a number line. Since distance is always non-negative, the absolute value is always positive or zero.

Consider \( |x| = 4 \):
  • This means the number \( x \) is exactly 4 units away from zero.
  • There are only two numbers on a number line that satisfy this condition: 4 and -4.
In real life, if you are told to count 4 steps away from a starting point (zero), you can go either in the positive direction to reach 4 or in the negative direction to reach -4. Both distances are the same.
solving equations
Solving absolute value equations involves understanding the definition of absolute value and its properties. Here's how to solve \( |x| = 4 \):

  1. Recognize that the absolute value equation \( |x| = 4 \) can be split into two separate linear equations because the distance of \( x \) from zero can be either 4 or -4.
  2. Set up the two possible equations: \( x = 4 \) and \( x = -4 \).
  3. Solve each equation separately:
    • From \( x = 4 \), we get \( x = 4 \).
    • From \( x = -4 \), we get \( x = -4 \).
By systematically setting up and solving these equations, you can find all possible values that satisfy the original absolute value equation.
positive and negative solutions
Absolute value equations often yield both positive and negative solutions because distance is independent of direction. For the equation \( |x| = 4 \):

  • The positive solution is \( x = 4 \) because the distance from zero to 4 is 4 units.
  • The negative solution is \( x = -4 \) because the distance from zero to -4 is also 4 units.

It's essential to verify both solutions by substituting back into the original equation:
  • For \( x = 4 \), \( |4| = 4 \), which satisfies the equation.
  • For \( x = -4 \), \( |-4| = 4 \), which also satisfies the equation.
This cross-check ensures that both solutions are indeed correct.

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Most popular questions from this chapter

When simplifying the expression \(3 x+4+2 x+7\) to \(5 x+11,\) some steps are usually done mentally. providing the property that justifies each statement in the given simplification. (These steps could be done in other orders.) $$ 3 x+4+2 x+7 $$ $$ =(3+2) x+(4+7) $$

Solve each problem. Kayla Koolbcck has $$ 37.60\( in her checking account. She uses her debit card to make purchases of \) 25.99\( and \) 19.34,\( which overdraws her account. Her bank charges her account an overdraft fee of \) 25.00 .\( She then deposits her paycheck for \) 58.66$ from her part-time job at Subway. What is the balance in her account?

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