Problem 71
Question
Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$\frac{x}{3}+2=\frac{x}{3}$$
Step-by-Step Solution
Verified Answer
The equation has no solution.
1Step 1: Subtract \(\frac{x}{3}\) from both sides to isolate the variable
Subtracting \(\frac{x}{3}\) from both sides of the equation \(\frac{x}{3} + 2 = \frac{x}{3}\), the left side becomes \(2\) and right side is \(0\). We could write the new equation as \(2 = 0\).
2Step 2: Identify whether the equation holds true
The equation \(2 = 0\) does not hold true as \(2\) is not equal to \(0\).
3Step 3: Interpretation of the result
Since the equation \(2 = 0\) is false, the original equation \(\frac{x}{3} + 2 = \frac{x}{3}\) has no solutions.
Key Concepts
Algebraic Equations No SolutionIsolate the Algebraic VariableAlgebraic Equation Interpretation
Algebraic Equations No Solution
Understanding when an algebraic equation has no solution is crucial in the study of algebra. An equation with no solution is one that, regardless of what value is substituted for the variable, will never be true. In the given exercise, after simplifying \(\frac{x}{3} + 2 = \frac{x}{3}\) by subtracting \(\frac{x}{3}\) from both sides, we are left with the equation \(2 = 0\).
This is a statement that is always false because the number 2 will never equal 0. In mathematical terms, we describe this situation by saying that the equation has no solution, or in set notation, we would write it as \(\emptyset\). The concept of an equation having no solution can be confusing, but it is simply an expression that results in a false statement, like \(2 = 0\), no matter what value you choose for the variable.
This is a statement that is always false because the number 2 will never equal 0. In mathematical terms, we describe this situation by saying that the equation has no solution, or in set notation, we would write it as \(\emptyset\). The concept of an equation having no solution can be confusing, but it is simply an expression that results in a false statement, like \(2 = 0\), no matter what value you choose for the variable.
Isolate the Algebraic Variable
Isolating the algebraic variable is a fundamental skill in solving equations. The objective is to get the variable on one side of the equation so you can determine what value makes the equation true. To isolate the variable, you perform operations that 'undo' what is being done to the variable, creating a simpler equivalent equation.
In our exercise, we attempted to isolate the variable \(x\) by subtracting \(\frac{x}{3}\) from both sides of \(\frac{x}{3} + 2 = \frac{x}{3}\). The idea was to have \(x\) on one side to facilitate solving. However, this resulted in the variable \(x\) being eliminated from the equation, leading to the constant equation \(2 = 0\). The act of attempting to isolate \(x\) unveiled that there's no value of \(x\) that will satisfy the original equation, hence, no solution exists.
In our exercise, we attempted to isolate the variable \(x\) by subtracting \(\frac{x}{3}\) from both sides of \(\frac{x}{3} + 2 = \frac{x}{3}\). The idea was to have \(x\) on one side to facilitate solving. However, this resulted in the variable \(x\) being eliminated from the equation, leading to the constant equation \(2 = 0\). The act of attempting to isolate \(x\) unveiled that there's no value of \(x\) that will satisfy the original equation, hence, no solution exists.
Algebraic Equation Interpretation
The final and perhaps most important step in solving equations is interpreting the meaning of the equation. After all the manipulation and simplification, what does the resulting statement tell us? Our example \(\frac{x}{3} + 2 = \frac{x}{3}\) led to \(2 = 0\) which is an impossibility.
The interpretation here is straightforward: since \(2 = 0\) is never true, the original equation does not balance for any real number value of \(x\). In mathematical terms, we articulate this conclusion by saying the solution set is empty (\emptyset). This interpretation step helps us understand the nature of the algebraic equation and its implications. In case of equations that do balance, interpreting the solution involves affirming the values of \(x\) that make the equation true, hence finding the solution set for the variable.
The interpretation here is straightforward: since \(2 = 0\) is never true, the original equation does not balance for any real number value of \(x\). In mathematical terms, we articulate this conclusion by saying the solution set is empty (\emptyset). This interpretation step helps us understand the nature of the algebraic equation and its implications. In case of equations that do balance, interpreting the solution involves affirming the values of \(x\) that make the equation true, hence finding the solution set for the variable.
Other exercises in this chapter
Problem 71
Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(3(2 y-1)
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Suppose that you put \(\$ 10,000\) in a rather risky investment recommended by your financial advisor. During the first year, your investment decreases by \(30
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Describe how volume is measured. Explain why linear or square units cannot be used.
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Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(4(2 y-1)>12\)
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