Problem 2 - Algebra 1: Concepts and Skills

Question

The solution of \(13 x=6\) rounded to the nearest hundredth is 0.46 Which of the following is a better way to list the solution? Explain. $$ \text { A. } x=0.46 $$ $$B. x ? 0.46$$

Step-by-Step Solution

Answer

Option B, \(x ? 0.46\), is a better way to represent the solution. This is because 0.46 is a rounded figure and x could be less than, greater than, or equal to 0.46.

Step 1: Understand the Meaning of the Equality

In mathematics, an equality (symbol =) denotes that two expressions represent the same quantity. Thus, when the given equation is solved, \(x = 0.46\) says that the variable x is exactly equal to 0.46.

Step 2: Understand the Meaning of the Inequality

On the other hand, an inequality (symbol ?) implies that one expression is not precisely equal, but rather less than or greater than another. So, \(x ? 0.46\) means that x is either less than, greater than, or equal to 0.46.

Step 3: Choose the Better Representation

Given that 0.46 is a rounded figure and not the exact solution of the equation, option B would be a more accurate representation. Despite x not being equal to the precise solution, it could be less than or greater than 0.46. Hence, considering the rounding off done to get the nearest hundredth, the better way to represent the solution is \(x ? 0.46\).

Key Concepts

Solving Equations

In algebra, solving equations is like solving a puzzle where you have to figure out the value of the unknown variable. The main goal is to find the number that makes the equation true. Each equation consists of certain components:

• Numbers: These are specific values you use in calculations.
• Variables: Symbols like \( x \) that represent unknown numbers.

When you solve an equation like \( 13x = 6 \), your task is to isolate the variable \( x \) on one side of the equation. By dividing both sides by 13, the solution becomes \( x = \frac{6}{13} \). This step simplifies to approximate \( x \approx 0.461538... \). The exact value of \( x \) is crucial because it tells us the precise quantity that satisfies the equation, but rounding can sometimes be used for simplicity.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify complex decimals by reducing the number of digits. It helps in making numbers easier to use and understand. The process of rounding involves several steps: In this specific exercise, we round 0.461538... to the nearest hundredth, which results in 0.46. Here is how it works:

• Find the place value you are rounding to (hundredths in this case).
• Look at the digit right after it (3rd decimal in this situation).
• If it is 5 or higher, round the previous digit up. If it is lower, keep it the same.

Rounding is useful when exact numbers are not necessary and a close approximate is sufficient, especially in real-world contexts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition or multiplication. Unlike equations, algebraic expressions do not include an equality sign. They are fundamental in algebra and can represent real-world situations.For example, in solving \( 13x = 6 \), before you isolate \( x \), it exists within an algebraic expression \( 13x \). Understanding how to manipulate these expressions is essential:

• Terms: These are parts of an expression separated by + or - signs.
• Coefficients: Numbers in front of variables, like 13 in \( 13x \).

The mastery of manipulating algebraic expressions is crucial for simplifying equations and finding solutions. This skill is key to constructing and solving more complex mathematical problems.