Problem 18 - Prealgebra

Question

For the given value, state whether each inequality is true or false. $$18-x>4, x=12$$

Step-by-Step Solution

Answer

The inequality \(18 - x > 4\), when \(x = 12\), is true.

Step 1: Substitute the Given Value

Replace the variable \(x\) with the given number 12 in the inequality. The inequality is \(18 - x > 4\). After substitution, it becomes \(18 - 12 > 4\).

Step 2: Compute the Left-Hand Side

Calculate the value on the left side of the inequality. Compute \(18 - 12 = 6\). Now the inequality is \(6 > 4\).

Step 3: Evaluate the Inequality

Compare the numbers on both sides of the inequality. Determine whether the inequality \(6 > 4\) is true or false. \(6\) is indeed greater than \(4\), so the inequality is true.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics, allowing us to work with numbers in a more dynamic way. They consist of numbers, variables (like \(x\)), and operators (such as +, -, *, /). An example of an algebraic expression is \(18 - x\). This particular expression implies that something is being subtracted from 18 using the variable \(x\). The power of algebraic expressions lies in their flexibility to accommodate different values for the variables.

• Variables represent unknown or changeable numbers.
• Operators specify the mathematical operations to perform.
• Understanding how to construct and deconstruct these expressions enables further exploration into solving equations and inequalities.

Substitution

Substitution is a crucial method in algebra to simplify expressions or equations by replacing variables with known values. This is useful to determine the truth of expressions or solve mathematical problems.
The process involves directly inserting the given number into the place of the variable. For instance, in substituting \(x = 12\) into the expression \(18 - x\), we replace \(x\) with 12 resulting in the expression \(18 - 12\).

• Substitution aids in simplifying the evaluation of expressions.
• It is a foundational concept in solving equations and inequalities.
• Ensure the value substituted corresponds to the same variable throughout the calculation.

Evaluating Inequalities

Evaluating inequalities involves determining if the statement made by the inequality is true or false. An inequality uses symbols like \(>\), \(<\), \(\leq\), \(\geq\) to compare values.
In our example, we look at the inequality \(6 > 4\), where 6 is the result of the left side of the equation (after substitution) and 4 is the constant on the right side.

• Start by simplifying both sides of the inequality as much as possible.
• Compare the two sides using the appropriate symbol.
• If the statement holds true after comparison, the inequality is true; otherwise, it is false.

Evaluating inequalities is essential for solving more complex algebraic problems and understanding mathematical relationships.