Problem 17 - Prealgebra

Question

Solve each inequality. Check your solution. $$3 m<33$$

Step-by-Step Solution

Answer

The solution to the inequality is \( m < 11 \).

Step 1: Divide Both Sides by 3

The inequality is given as \( 3m < 33 \). To solve for \( m \), divide every term of the inequality by 3: \begin{align*} \frac{3m}{3} &< \frac{33}{3} \ m &< 11 \\end{align*} Now, we have the inequality \( m < 11 \).

Step 2: Check the Solution

To ensure the solution is correct, choose a test value for \( m \) that satisfies \( m < 11 \). Let's select \( m = 10 \). Substitute it back into the original inequality: \begin{align*} 3(10) &< 33 \ 30 &< 33 \\end{align*} The inequality holds true, confirming that the solution \( m < 11 \) is correct.

Key Concepts

Prealgebra

Prealgebra serves as the foundation for all future math learning, especially when it comes to algebra. It introduces students to basic math concepts, including numbers, simple operations, and eventually to more complex ideas like variables and inequalities. The ultimate goal of prealgebra is to prepare students for algebra.
Prealgebra often involves:

• Understanding numbers and operations such as addition, subtraction, multiplication, and division.
• Learning about fractions, decimals, and percentages.
• Starting with variables and simple algebraic expressions.
• Introduction to basic geometry concepts.

The inequality exercise presented, such as solving \(3m < 33\), is a perfect example of prealgebra content as it incorporates variables and requires the application of basic mathematical operations to solve.
This ensures students grasp how to manipulate and solve equations and inequalities as a preparation for algebra.

Solving Inequalities

Solving inequalities is a key component in mathematics, particularly when moving from arithmetic to algebra. An inequality shows a relationship between expressions that are not equal. In this context, it is essential to understand how to solve for a variable that satisfies a given condition.
The process generally involves:

• Identifying the inequality and what you need to solve for.
• Performing the same operations on both sides of the inequality sign. This often involves operations like addition, subtraction, multiplication, or division, as seen in the \(3m < 33\) example where dividing both sides by 3 was crucial.
• Maintaining the inequality direction. This is important, especially when multiplying or dividing both sides by a negative number, as the inequality sign would need to be flipped.

After performing these steps, it's important to check the solution by substituting a value back into the original inequality, ensuring it holds true. This verifies the accuracy of the solution.

Mathematical Reasoning

Mathematical reasoning is the logical framework in which we approach math problems. It is not just about how to perform calculations, but about understanding why certain operations are performed and what they achieve. This capability is crucial when solving prealgebra problems and inequalities.
The skill of mathematical reasoning involves:

• Evaluating given problems and determining what needs to be found or solved.
• Breaking down complex problems into manageable steps.
• Justifying and explaining solutions and processes. For example, in solving \(3m < 33\), reasoning tells us to isolate \(m\) by dividing by 3.
• Reflecting on the results by checking the solution.

Mathematical reasoning helps students to better understand problems and solutions, thus enabling them to tackle more complicated math in the future. It's about building a solid framework of logic and problem-solving skills that are applied in mathematics.