Problem 27 - Algebra 2

Question

Define a variable and write an inequality for each problem. Then solve. The product of 12 and a number is greater than 36

Step-by-Step Solution

Answer

\( x > 3 \)

Step 1: Define the Variable

Let's define the variable that will represent the unknown number. Let \( x \) be the unknown number we are trying to find.

Step 2: Write the Inequality

We are given that "the product of 12 and a number is greater than 36". To express this in an inequality, we write:\[ 12x > 36 \] This inequality states that 12 times \( x \) is greater than 36.

Step 3: Solve the Inequality

To isolate \( x \), we need to divide both sides of the inequality by 12:\[ \frac{12x}{12} > \frac{36}{12} \]Simplifying both sides, we get:\[ x > 3 \] So, \( x \) must be greater than 3 to satisfy the inequality.

Key Concepts

Variable Definition

Defining a variable is one of the first steps in solving algebraic problems. A variable can be thought of as a placeholder for a number that we do not yet know. In math, we often use letters like \( x \), \( y \), or \( z \) as variables.
For instance, in our problem, the unknown number needs a symbol to represent it. This makes it easier to describe mathematical relationships. We chose \( x \) for our variable. This means wherever we see \( x \), it is referring to that unknown number.
Think of it as a nametag for a number that you'll discover later. Variables are essential for writing both equations and inequalities because they allow us to generalize and solve problems systematically.

Algebraic Expressions

Algebraic expressions use variables to describe mathematical relationships in a general form. These expressions can include numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.
In our exercise, the expression "the product of 12 and a number" refers to 12 times our unknown variable, which becomes \( 12x \). This expression forms the core part of our inequality.
Algebraic expressions help simplify the problem-solving process, enabling you to manipulate the expression and see how changes affect the whole problem. In math, these expressions are crucial, as they form the building blocks for writing equations and inequalities.

Solving Inequalities

Solving inequalities involves finding the value or range of values that make the inequality true. An inequality, unlike an equation, does not necessarily have a single solution. Instead, it shows a spectrum of solutions that meet the criteria.
For the inequality \( 12x > 36 \), it indicates that the product of 12 and our variable \( x \) must be greater than 36. To solve this, we divide both sides by 12 to find \( x \).

• Divide each side of the inequality by the same positive number to preserve the direction of the inequality: \( \frac{12x}{12} > \frac{36}{12} \).
• This simplification leads to \( x > 3 \).

This tells us that \( x \) must be any number greater than 3 to satisfy the inequality. Solving inequalities requires careful manipulation of both sides while keeping track of the inequality's direction.