Question

Solve the equation. Round the result to the nearest tenth if necessary. $$5(2 x+2.3)-11.2=6 x-5$$

Short answer

The solution to the equation is \(x=-1.3\) when rounded to the nearest tenth.

Step-by-step solution

Distribute the multiplication

First, apply the distribution property of multiplication over addition to the left side of the equation: \(5(2x+2.3)=10x+11.5\). So, the equation becomes: \[10x+11.5-11.2=6x-5\]

Combine like terms

Next, combine like terms and get: \[10x+0.3=6x-5\].

Isolate the variable x

Isolate the variable x by removing 6x from both sides. The equation now turns into \[4x+0.3=-5\]. Then remove 0.3 from both sides to get \[4x=-5.3\] on continuing, to isolate x, divide each side by 4. Therefore, \[x=-5.3/4\]

Calculate the final answer

After calculating, it is found that \[x=-1.325\]. Since the question asks for the result to be rounded to the nearest tenth, the solution to the equation would be \[x=-1.3\]

Key concepts

Distribution Property

Understanding the distribution property is essential when solving linear equations as it allows us to simplify expressions and make them easier to manage. The distribution property states that when you multiply a sum or difference by a number, you must multiply each term inside the parentheses by that number. In our exercise, we apply this property to the expression \(5(2x+2.3)\). This means we need to multiply both \(2x\) and \(2.3\) by \(5\) separately, which gives us \(10x+11.5\).

Applying the distribution property correctly is the first step towards simplifying the equation and getting it ready for further operations, such as combining like terms and isolating the variable.

Combining Like Terms

Combining like terms is a process that helps to further simplify the equation. Like terms are terms that have the same variables raised to the same power. In this context, we focus on combining terms with the variable \(x\) and constant numbers (without a variable). Our equation, after distribution, includes the terms \(10x\) and \(6x\) which are like terms, as well as \(11.5\) and \(11.2\), which are constants and hence like terms.

By subtracting \(11.2\) from \(11.5\), we combine the constants, which results in \(0.3\). By this step, we've simplified the equation from \(10x+11.5-11.2\) to \(10x+0.3\), making it cleaner and preparing the stage to isolate the variable.

Isolate the Variable

To solve for \(x\), we need to isolate the variable. This means we need to get \(x\) alone on one side of the equation. After combining like terms, we subtract \(6x\) from both sides of the equation \(10x+0.3=6x-5\) to remove it from the right side. This leaves us with \(4x+0.3=-5\).

The next step is to get rid of the \(0.3\) by subtracting it from both sides, resulting in \(4x=-5.3\). Finally, to completely isolate \(x\), we divide both sides by \(4\), leading to the isolated variable \(x\) on one side of the equation: \(x=-5.3/4\). Isolating the variable is a critical step in solving any algebraic equation, as it gives us the value of the unknown that we're solving for.

Rounding Numbers

Rounding numbers is an important concept, especially when dealing with precise measurements or when an exact answer isn't necessary. When we solve the equation \(x=-5.3/4\), we get \(x=-1.325\). However, the exercise requires us to round the result to the nearest tenth. To round a number to the nearest tenth, we look at the hundredths place; if it's \(5\) or above, we round up, and if it's below \(5\), we round down.

In this case, \(5\) in the hundredths place tells us to round up the tenth's place from \(2\) to \(3\). Thus, the final rounded answer is \(x=-1.3\). It’s crucial to know when and how to round numbers, as it can affect the accuracy of your final answer in real-world applications.