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Chapter 6: Problem 77

Solve each equation. Use set notation to express solution sets for equations with no solution or equations that are true for all real numbers. \(7+2(3 x-5)=8-3(2 x+1)\)

Short Answer

Expert verified
The solution set of the equation is the empty set, written in set notation as \(\emptyset\).

Step by step solution

01

Simplify the Equation

First, apply the distributive property on both sides of the equation, i.e multiply the numbers outside the brackets with each term within the bracket. This results in 7 + 6x -10 = 8 - 6x -3. Deleting the brackets gives 6x - 3 = -6x + 5.
02

Transposition of terms

To gather like terms, move -6x from the right side of the equation to the left side, which results in 6x - 6x = 3 + 5. Combine like terms and simplify to obtain 0=8.
03

Finalize Answer

The simplified equation 0=8 is not true. Therefore, it implies that the original equation has no solution.

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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 a key concept in algebra that helps simplify complex expressions. When you encounter an expression like \(a(b + c)\), the distributive property allows you to expand it into \(ab + ac\). It's like spreading out the multiplication over addition or subtraction. This principle holds for any number of terms inside the parentheses.
  • You multiply the term outside the parentheses by each term inside the parentheses.
  • This is essential for removing parentheses in equations.
  • The operation is often used to simplify expressions before solving equations.
In our example, we apply this property to the equation \(7+2(3x-5)=8-3(2x+1)\). This means we distribute the \(2\) across \((3x-5)\) to get \(6x - 10\), and distribute \(-3\) across \((2x+1)\) to get \(-6x - 3\). It's a critical first step as it helps remove the parentheses, enabling further simplification.
Transposition of Terms
Transposition is a strategy that involves moving terms from one side of an equation to the other to isolate a specific term or simplify the equation. Think of it as balancing the equation by keeping it equal on both sides which is essential for solving for a variable.
  • If you move a term to the other side, you reverse its operation. For example, \(+5\) becomes \(-5\) when transposed.
  • The primary goal is often to gather like terms together.
  • This method helps keep track of changes to equation structure.
In our exercised equation, after using the distributive property, we have \(6x - 3 = -6x + 5\). To simplify, we move \(-6x\) from the right to the left side, resulting in \(6x - (-6x) = 3 + 5\), which simplifies to \(0 = 8\). This rearrangement helps in combining like terms effectively.
No Solution in Equations
An equation resulting in a mathematical impossibility, such as contradictory statements, helps us identify equations with no solution. This happens when the final simplified form does not hold as a true statement. The result of the rearrangement and simplification might sometimes lead to an incorrect equation like \(0 = 8\).
  • If the equation simplifies to something impossible (e.g., \(0 = 5\)), then there is no solution.
  • It implies that no real number can satisfy the original equation.
  • This is conveyed in set notation as \(\emptyset\), indicating an empty set with no numbers to solve the equation.
In our example, simplifying leads to \(0 = 8\), making it clear that there isn鈥檛 any real value of \(x\) that can satisfy the initial equation. Understanding such results is crucial for recognizing when equations simply have no solutions.

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Most popular questions from this chapter

Despite booming new car sales with their cha-ching sounds, the average age of vehicles on U.S. roads is not going down. The bar graph shows the average price of new cars in the United States and the average age of cars on U.S. roads for two selected years. Exercises 23-24 are based on the information displayed by the graph. In 2014 , the average price of a new car was \(\$ 37,600\). For the period shown, new-car prices increased by approximately \(\$ 1250\) per year. If this trend continues, how many years after 2014 will the price of a new car average \(\$ 46,350\) ? In which year will this occur?

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(3 x^{2}+7 x+2\)

It was wartime when the Ricardos found out Mrs. Ricardo was pregnant. Ricky Ricardo was drafted and made out a will, deciding that \(\$ 14,000\) in a savings account was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Ricky stipulated that if the child were a boy, he would get twice the amount of the mother's portion. If it were a girl, the mother would get twice the amount the girl was to receive. We'll never know what Ricky was thinking of, for (as fate would have it) he did not return from the war. Mrs. Ricardo gave birth to twins \(-\) a boy and a girl. How was the money divided?

In Exercises 71-76, use set-builder notation to describe all real numbers satisfying the given conditions. A number increased by 5 is at least two times the number.

Solve each proportion and check. \(\frac{2}{y-5}=\frac{3}{y+6}\)
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