/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 115 $$\text { Solve } 3-5|x-4|=0$$... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 115

$$\text { Solve } 3-5|x-4|=0$$

Short Answer

Expert verified
x = 23/5 or x = 17/5.

Step by step solution

01

Isolate the absolute value term

Start by moving the constant term on one side of the equation to simplify it. The original equation is 3 - 5|x-4| = 0. Subtract 3 from both sides to isolate the absolute value term: 3 - 5|x-4| - 3 = 0 - 3, which simplifies to: -5|x-4| = -3.
02

Solve for the absolute value expression

Divide both sides of the equation by -5 to get the absolute value term by itself: -5|x-4| / -5 = -3 / -5. This simplifies to: |x-4| = 3/5.
03

Split into two cases

Since we have the absolute value isolated, |x-4| = 3/5 means there are two cases to consider: Case 1: x - 4 = 3/5 Case 2: x - 4 = -3/5.
04

Solve for x in each case

Solve each equation separately: Case 1: x - 4 = 3/5. Add 4 to both sides: x = 4 + 3/5 x = 20/5 + 3/5 = 23/5. Case 2: x - 4 = -3/5. Add 4 to both sides: x = 4 - 3/5 x = 20/5 - 3/5 = 17/5.

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Key Concepts

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

Understanding Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. For instance, the absolute value of both 3 and -3 is 3, written as \(|3| = 3\) and \(|-3| = 3\). Absolute values are always non-negative.
When dealing with absolute value equations, we're essentially splitting one equation into two possible equations because the inside of the absolute value can be both positive and negative.
In our exercise, \(|x - 4| = \frac{3}{5}\), it means the term \(x-4\) can either equal \(\frac{3}{5}\) (the positive case) or \(-\frac{3}{5}\) (the negative case). This dual nature comes from the definition of absolute value.
Isolating Terms
Before solving the absolute value equation, it's crucial to isolate the absolute value term. In our example, we start with the equation:
\(3 - 5|x-4| = 0\).
To isolate \(|x-4|\), follow these steps:
  • Subtract 3 from both sides: \(-5|x-4| = -3\).
  • Divide both sides by -5: \(|x-4| = \frac{3}{5}\).
By isolating the absolute value term, we simplify the problem and can now handle the two resulting cases separately.
Solving Equations with Absolute Value
Once the absolute value term is isolated, split the problem into two linear equations:
  • \(x - 4 = \frac{3}{5}\)
  • and \(x - 4 = -\frac{3}{5}\)
Now solve for \(x\) in each case:

First Case (positive):
\(x - 4 = \frac{3}{5}\)
Add 4 to both sides:
\(x = 4 + \frac{3}{5} = 4 + 0.6 = 4.6 = \frac{23}{5}\)

Second Case (negative):
\(x - 4 = -\frac{3}{5}\)
Add 4 to both sides:
\(x = 4 - \frac{3}{5} = 4 - 0.6 = 3.4 = \frac{17}{5}\)

Combining these, the solutions are \(x = \frac{23}{5} \) and \(x = \frac{17}{5}\).

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Most popular questions from this chapter

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