Problem 53 - Prealgebra

Question

ALGEBRA For the given value, state whether each inequality is true or false. (Lesson 8-3) $$\frac{d}{2} \geq 8, d=4$$

Step-by-Step Solution

Answer

False, 2 is not greater than or equal to 8.

Step 1: Substitute the Given Value

First, replace the variable \(d\) in the inequality \(\frac{d}{2} \geq 8\) with the provided value \(d = 4\). This yields the expression \(\frac{4}{2} \geq 8\).

Step 2: Simplify the Expression

Calculate the left side of the inequality. Dividing the numerator by the denominator gives \(\frac{4}{2} = 2\). The inequality now is \(2 \geq 8\).

Step 3: Evaluate the Inequality

Compare the simplified left side (2) with the right side (8) in the inequality \(2 \geq 8\). Since 2 is not greater than or equal to 8, the inequality is false.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. Typically, expressions help us represent real-world situations in a concise form or solve mathematical problems.
Variables are letters that stand for unknown or changeable values. In the given inequality, \( \frac{d}{2} \geq 8 \), the variable \( d \) represents a number that you can substitute with specific values. This makes algebra incredibly versatile and powerful for solving a wide range of problems.
Algebraic expressions can be simplified by performing the operations indicated and combining like terms. For instance, simplifying \( \frac{4}{2} \) results in \( 2 \). Simplification is an essential skill in algebra as it makes solving equations and inequalities straightforward.

Substitution Method

The substitution method in algebra involves replacing a variable with its given value in an expression or equation. This process aims to transform an expression with variables into one with concrete numbers that can be easily simplified or evaluated.
In the problem \( \frac{d}{2} \geq 8 \), using the substitution method means replacing \( d \) with 4 to form the expression \( \frac{4}{2} \geq 8 \).

• Begin by identifying the variable in the equation or inequality.
• Plug the given number where the variable appears.
• Simplify the expression further to check for correctness or solve the given mathematical task.

Understanding substitution ensures you can translate algebraic expressions into simpler numerical forms, aiding in logical problem-solving processes.

Evaluating Inequalities

Evaluating inequalities involves determining whether the inequality statement is true or false given specific values. It requires comparing the expressions on either side of the inequality sign.
In the example \( 2 \geq 8 \), evaluation involves checking if the left side (2) is greater than or equal to the right side (8). Clearly, 2 is less than 8, so the statement is false. Here’s a step-by-step way to evaluate similar problems:

• First, ensure the expression is simplified.
• Next, substitute any given values for variables.
• Then, assess whether the simplified expression meets the inequality conditions.

Crucially, accurately evaluating inequalities helps us make sound conclusions about mathematical relationships and conditions.