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Chapter 2: Problem 77

Solve each equation. $$5 x-2(3 x+6)=4-(2+x)+7$$

Short Answer

Expert verified
There is no solution to the equation.

Step by step solution

01

- Simplify inside the parentheses

Simplify the expression within the parentheses first: - The left side of the equation: a. Distribute the \(-2\) through the \( (3x + 6) \) \(5x - 2(3x + 6) = 5x - 6x - 12\)- The right side of the equation:a. Combine like terms: \(4 - 2 - x + 7 = 9 - x\)So, now we have the equation: \(5x - 6x - 12 = 9 - x\)
02

- Combine like terms

Combine the \(5x\) and \(-6x\) terms on the left side: \(5x - 6x - 12 = 9 - x\) This simplifies to: \(- x - 12 = 9 - x\)
03

- Add \(x\) to both sides

Add \(x\) to both sides of the equation to eliminate the variable from one side: \(-x + x - 12 = 9 - x + x\) This simplifies to: \(- 12 = 9\)
04

- Combine constants

Observe that the previous equation cannot be true as \(-12\) is never equal to \(9\). Therefore, there is no solution to the equation.

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Key Concepts

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

Distributive Property
The distributive property is fundamental in solving linear equations. It allows you to multiply a single term by each term inside a set of parentheses. In this exercise, the equation given is: \[5x - 2(3x + 6) = 4 - (2 + x) + 7.\] First, apply the distributive property to \(-2(3x + 6)\): \[5x - 6x - 12.\] Here, \(-2\) multiplied by \(3x\) and \(-2\) multiplied by \(6\) results in \(-6x\) and \(-12\). This step is crucial because it helps to open up the grouping and makes it easier to move on to combining like terms.
If you overlook the distributive property, it can lead to incorrect simplifications.
Combining Like Terms
Combining like terms simplifies expressions by grouping and summing terms that have common variables or constants. In our example, after applying the distributive property, the equation becomes: \[5x - 6x - 12 = 9 - x.\] You need to combine the like terms on both sides. On the left side:
\[5x - 6x\] - combine similar terms to get \(-x\): \[- x - 12 = 9 - x.\] Combining like terms reduces the complexity of the equation, and it becomes easier to identify if there's a simpler form or inherent inconsistency that may indicate no solution.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. After combining like terms in our equation, it鈥檚 essential to simplify further. We have: \[- x - 12 = 9 - x.\] Next, add x to both sides of the equation: \[- x + x - 12 = 9 - x + x.\] This step effectively cancels the variable term on both sides, leaving us with: \[- 12 = 9.\] Simplifying correctly helps you identify obvious contradictions or verify steps accurately, ensuring an accurate solution to the problem.
No Solution in Equations
Sometimes, after simplifying an equation, you might end up with a statement that isn鈥檛 true, indicating there's no solution. In our case: \[- 12 = 9.\] Since \-12\ is never equal to \9\, there's no value for \x\ that can satisfy the equation. It鈥檚 important to recognize these contradictions because it means the initial equation has no valid solution. This can occur when both sides of the equation simplify to different constants, highlighting an inherent inconsistency in the provided equation. Recognizing such a scenario is an essential part of mastering equation-solving skills.

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Most popular questions from this chapter

Solve problem by using an inequality. Car shopping. Jennifer is shopping for a new car. In addition to the price of the car, there is an \(8 \%\) sales tax and a \(\$ 172\) title and license fee. If Jennifer decides that she will spend less than \(\$ 10,000\) total, then what is the price range for the car?

Show a complete solution to each problem. Really odd integers. If the smaller of two consecutive odd integers is subtracted from twice the larger one, then the result is \(13 .\) Find the integers.

Show a complete solution to each problem. GE profit. General Electric posted a third quarter profit of 44 cents per share. This profit was \(15.8 \%\) greater than the third quarter profit of the previous year. What was the profit per share in the third quarter of the previous year?

Show a complete solution to each problem. Financial independence. Dorothy had \(\$ 8000\) to invest. She invested part of it in an investment paying \(6 \%\) and the rest in an investment paying \(9 \% .\) If the total income from these investments was \(\$ 690\), then how much did she invest at each rate?

Solve each problem by using a compound inequality. See Example 12. Selling-price range. Renee wants to sell her car through a broker who charges a commission of \(10 \%\) of the selling price. The book value of the car is \(\$ 14,900,\) but Renee still owes \(\$ 13,104\) on it. Although the car is in only fair condition and will not sell for more than the book value, Renee must get enough to at least pay off the loan. What is the range of the selling price?
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