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Chapter 1: Problem 105

What Is Wrong? One step in the following list contains an error. Identify it and explain what is wrong. $$ \begin{aligned} x &=2 \\ 3 x-2 x &=2 \\ 3 x &=2 x+2 \\ x^{2}+3 x &=x^{2}+2 x+2 \\ x^{2}+3 x-10 &=x^{2}+2 x-8 \\ (x-2)(x+5) &=(x-2)(x+4) \\ x+5 &=x+4 \\ 1 &=0 \end{aligned} $$

Short Answer

Expert verified
The error is at Step 5. The subtraction resulted in an incorrect equation leading to a false statement.

Step by step solution

01

Understand the Given Problem

The problem involves an algebraic error in the following equations. The goal is to identify and explain the mistake.
02

Identify Each Equation

List the given equations to observe their progression:1. \( x = 2 2. 3x - 2x = 2 \)3. \( 3x = 2x + 2 \)4. \( x^2 + 3x = x^2 + 2x + 2 \)5. \( x^2 + 3x - 10 = x^2 + 2x - 8 \)6. \( (x-2)(x+5) = (x-2)(x+4) \)7. \( x+5 = x+4 \)8. \( 1 = 0 \)
03

Analyze Each Step for Consistency

Check every step to make sure it logically follows from the previous step.
04

Detail Examination of Steps

Let's start reviewing each step in detail:- From Step 1: \( x = 2 \)- Step 2: Simplifies correctly to \( 3x - 2x = 2 \), which is \( x = 2 \).- Step 3: \( 3x = 2x + 2 \) follows from the previous step by substituting \( x = 2 \).- Step 4: \( x^2 + 3x = x^2 + 2x + 2 \) by adding \( x^2 \) to both sides.- Step 5: \( x^2 + 3x - 10 = x^2 + 2x - 8 \) is incorrect.
05

Identify and Explain the Error

The error occurs between Step 4 and Step 5. Step 4 gave \( x^2 + 3x = x^2 + 2x + 2 \). Subtracting \( x^2 \) from both sides results in \( 3x = 2x + 2 \), not the given steps in the problem.
06

Correct Step 5

Correct the equation to resolve: \( 3x - 2x = 2 \) which simplifies to \( x = 2 \)
07

Revised Steps

To accurately solve the system without error, the equations should instead follow:1. \( x = 2 \)2. \( 3x - 2x = 2 \)3. \( x = 2 \)

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Key Concepts

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

Error Identification
Spotting errors in algebra is crucial for solving equations correctly. It helps you ensure the steps you are taking are logical and mathematically sound. In the given exercise, we encountered a critical error that disrupted the whole calculation. The error was between Step 4 and Step 5.
Let's briefly go over why this happened:
- In Step 4, we have: \( x^2 + 3x = x^2 + 2x + 2 \). This step is correct.
- But in Step 5: \( x^2 + 3x -10 = x^2 + 2x - 8 \). This transformation doesn鈥檛 follow logically from Step 4.

To identify such errors:
  • Always compare each step with the previous ones to ensure consistency.
  • Spot any abrupt or illogical jumps in the equation progression.
Knowing where and why the mistake occurred allows us to correct it and prevent similar errors in the future.
Equation Consistency
Maintaining consistency throughout the steps of solving an equation is essential for accurate results. Consistency ensures that each transformation is valid and that we're following algebraic rules correctly.
Here are some key points to keep in mind:
  • Each new step should clearly follow from the one before.
  • Any changes made to one side of the equation must be made to the other side.
  • Watch for any signs that logic might be disrupted, such as contradictions or impossible results (like 1 = 0).

On analyzing the exercise:
- Step 4 gave us: \( x^2 + 3x = x^2 + 2x + 2 \)
- From here, subtracting \( x^2 \) from both sides simplifies to: \( 3x = 2x + 2 \)
-But in step 5, this wasn鈥檛 followed, leading to the error.
Properly maintaining equation consistency avoids such pitfalls.
Algebraic Simplification
Simplification is a fundamental part of solving algebraic equations. Simplifying means expressing the equation in its simplest form without changing its value. It makes the problem easier to solve and understand.

In the given exercise, proper simplification goes hand in hand with maintaining consistency. We look to transform equations in valid ways that make solving straightforward:
Consider the equations:
- Step 1: \( x = 2 \)
- Steps 2 and 3 simplify correctly to: \( 3x - 2x = 2 \) and \( 3x = 2x + 2 \)
- Step 4 further simplifies correctly, adding \( x^2 \) to get: \( x^2 + 3x = x^2 + 2x + 2 \)
- Then, step 5 should simplify by subtracting \( x^2 \) from both sides and solving for x: \( 3x = 2x +2 \).

Key tips for algebraic simplification:
  • Combine like terms wherever possible.
  • Make use of operations that simplify the equation logically.
  • Be mindful of each operation's validity and impact on the equation's balance.

By understanding and correctly applying simplification, the solutions become clearer and more manageable.

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Most popular questions from this chapter

Specifications for a rod in an internal combustion engine call for a length of 5.375 inches. Lengths within 0.0025 inch of this length are acceptable. Express this situation as an inequality involving an absolute value. Use \(x\) as the actual rod length and solve for \(x\).

High-Speed Walkways Toronto's Pearson International Airport has a high-speed version of a moving walkway. If Liam walks while riding this moving walkway, he can travel 280 meters in 60 seconds less time than if he stands still on the moving walkway. If Liam walks at a normal rate of 1.5 meters per second, what is the speed of the walkway?

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

In the 2016 Olympics, Usain Bolt of Jamaica won the gold medal in the 100 -meter race with a time of 9.81 seconds. In the 1896 Olympics, Thomas Burke of the United States won the gold medal in the 100-meter race in 12.0 seconds. If they ran in the same race, repeating their respective times, by how many meters would Bolt beat Burke?

True or False Radical equations sometimes have extraneous solutions.
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