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Chapter 2: Problem 76

For exercises 65-86, (a) solve. (b) check. $$ 10 u-(u+4)=9 u-2 $$

Short Answer

Expert verified
No solution

Step by step solution

01

Distribute the terms within parentheses

Expand the equation by distributing the negative sign inside the parentheses: \[ 10u - (u + 4) = 9u - 2 \]Which simplifies to: \[ 10u - u - 4 = 9u - 2 \]
02

Combine like terms

Combine the terms involving \(u\) on the left side: \[ (10u - u) - 4 = 9u - 2 \]This simplifies to: \[ 9u - 4 = 9u - 2 \]
03

Move all terms involving \(u\) to one side

Subtract \(9u\) from both sides of the equation: \[ 9u - 4 - 9u = 9u - 2 - 9u \]This simplifies to: \[ -4 = -2 \]
04

Analyze the resulting statement

The equation \( -4 = -2 \) is false. This indicates that there are no values of \(u\) that can satisfy the original 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 a key tool in algebra. It lets you multiply a single term by each term inside parentheses. For the equation given in the exercise, we need to distribute a negative sign. The original equation is:

\[10u - (u + 4) = 9u - 2\]

When we apply the distributive property, we distribute the negative sign to both terms inside the parentheses. This means we multiply -1 by each term:

\[10u - u - 4 = 9u - 2\]

Notice how the negative sign affects both the \(u\) and the \(4\). This step is crucial because it simplifies the equation and eases further steps.

Always be careful with distributing negative signs. It鈥檚 a common area for mistakes!
Like Terms
Likewise, when solving algebraic equations, combining like terms is another important step. Like terms are terms that have the same variable raised to the same power. In our problem, both \(10u\) and \(u\) are like terms because they both have the variable \(u\). Let鈥檚 look back at our exercise:

\[10u - u - 4 = 9u - 2\]

To combine like terms, you sum up the coefficients (numbers in front of the variables). For our equation:

\[10u - u\]

Which simplifies to:

\[9u - 4 = 9u - 2\]

After combining like terms, you can see we鈥檝e reduced the complexity of the equation. This makes it easier to move terms around and subsequently solve the equation.
No Solution
Finally, the concept of 'no solution' is critical in algebra. It occurs when an equation simplifies to a statement that is always false. In our problem, after moving all the like terms involving \(u\) to one side, we got:

\[-4 = -2\]

This is a contradiction because -4 is never equal to -2. When this happens, it means there are no possible values of \(u\) that will make the equation true.


You might encounter similar equations in your algebra journey. Recognizing a 'no solution' scenario saves time and helps understand deeper properties of equations. Always check and analyze the resulting statement to conclude whether the original equation has solutions or not.

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Most popular questions from this chapter

Solve \(P=\frac{1}{3} b h\) for \(h\).

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(-2 x+12>36\) Incorrect Answer: \(-2 x+12>36\) $$ \begin{aligned} -12 &-12 \\ \hline-2 x+0 &>24 \\ \frac{-2 x}{-2} &>\frac{24}{-2} \\ x &>-12 \end{aligned} $$

Find the enrollment at Oregon State University before the increase. Round to the nearest whole number. Oregon State University saw the biggest growth, a \(5.3\) percent jump of 1,302 students. Portland State University grew by only \(1.5\) percent, but remains the state's largest university with 28,958 students. (Source: www.oregonlive .com, Nov. 10. 2011)

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ 41 \leq-5 h+40 $$

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