Problem 30 - Prealgebra

Question

Four times a number less 6 is greater than two times the same number plus \(8 .\) For what number or numbers is this true?

Step-by-Step Solution

Answer

The inequality is true for numbers greater than 7.

Step 1: Define the Variable

Let's use the variable \(x\) to represent the unknown number we are trying to find.

Step 2: Formulate the Inequality

Translate the word problem into an inequality. The phrase "four times a number less 6" can be written as \(4x - 6\). The phrase "two times the same number plus 8" can be written as \(2x + 8\). According to the problem, "four times a number less 6 is greater than two times the same number plus 8" can be formulated as: \ \[ 4x - 6 > 2x + 8 \]

Step 3: Solve the Inequality

Subtract \(2x\) from both sides to begin isolating \(x\): \ \[ 4x - 2x - 6 > 8 \] Simplify it to get: \ \[ 2x - 6 > 8 \]

Step 4: Isolate the Variable

Add \(6\) to both sides to further isolate \(2x\): \ \[ 2x - 6 + 6 > 8 + 6 \] Simplify to obtain: \ \[ 2x > 14 \]

Step 5: Solve for x

Divide both sides of the inequality by \(2\) to solve for \(x\): \ \[ x > 7 \]

Step 6: Interpret the Solution

This inequality solution tells us that any number greater than 7 will satisfy the original condition set by the problem.

Key Concepts

Variable Representation

When approaching word problems in mathematics, it's crucial to start by identifying the variables. A variable is a symbol, most commonly represented by letters like \(x\), \(y\), or \(z\), that stands for an unknown or changing quantity. In our example problem, we aim to find a number that meets certain conditions. The best step is to use the letter \(x\) as our variable symbol.

Variable representation serves as a foundation for creating equations and inequalities that describe the situation in the problem. By defining the variable carefully, you establish the groundwork needed to translate complex worded statements into manageable mathematical expressions. As a rule of thumb, choose a clear and consistent variable to avoid confusion later in your calculations.

Always remember:

• Pick a symbol for the unknown quantity.
• Ensure your variable is defined in the context of the problem.
• Remain consistent with this variable throughout your solution.

Solving Inequalities

Once a variable and the inequality have been established, the next task is to solve it by isolating the variable on one side. Inequalities express relationships between expressions that are not necessarily equal, instead using signs like \(>\), \(<\), \(\geq\), or \(\leq\). In our example, the inequality is \(4x - 6 > 2x + 8\).

Here are some steps to solve an inequality:

• First, simplify each side of the inequality separately if needed.
• Bring like terms to one side—subtract or add terms to isolate the variable.
• The objective is to have the inequality in a form \(x > a\) or \(x < a\), where \(a\) is a constant.

This method reveals the set of values that satisfy the inequality condition. Remember, when multiplying or dividing by a negative number, the direction of the inequality sign flips.

In our exercise, solving \(4x - 6 > 2x + 8\) eventually led us to \(x > 7\), indicating all numbers greater than 7 satisfy the condition.

Translating Word Problems into Math Expressions

The ability to turn a complex word problem into a tidy mathematical expression is a critical skill. This process involves discerning the operations and relationships described in words and converting them into symbols and formulas. Our task was to handle the statement: "Four times a number less 6 is greater than two times the same number plus 8."

Here's how you can approach this:

• Identify operations described in the problem. For instance, "less" signifies subtraction, "times" denotes multiplication, etc.
• Convert these into numerical and algebraic expressions, like \(4x - 6\) or \(2x + 8\).
• Combine these expressions into an inequality or equation based on the context.

This formula creation helps simplify the problem-solving process, allowing you to focus on solving the math rather than decoding the language. Translating accurately ensures the solution reflects the original problem's requirements. Working through this example, the inequality \(4x - 6 > 2x + 8\) directly represents the conditions laid out in the word problem.