Problem 19 - Prealgebra

Question

For the given value, state whether each inequality is true or false. $$14+n<23, n=8$$

Step-by-Step Solution

Answer

The inequality is true.

Step 1: Substitute the Value

First, take the given equation \(14 + n < 23\) and substitute \(n = 8\) into the inequality. This transforms the inequality into \(14 + 8 < 23\).

Step 2: Perform the Addition

Calculate the left side of the inequality: \(14 + 8 = 22\). Now the inequality becomes \(22 < 23\).

Step 3: Compare the Results

Compare the two quantities involved in the inequality \(22 < 23\). Since 22 is indeed less than 23, the inequality holds true.

Key Concepts

Substitution

Substitution is a technique where you replace a variable with a given numerical value. This method simplifies equations, making them easier to solve. In the context of inequalities, substitution involves replacing the variable in the inequality with a specified value to determine the resulting expression. For example, in our given problem, the inequality is initially written as \(14 + n < 23\). The value specified for \(n\) is 8. When we substitute, we replace every occurrence of \(n\) in the inequality with the number 8. After substitution, the inequality becomes \(14 + 8 < 23\). This step is crucial as it sets the stage for further calculations and analysis.

Comparison

Comparison is the step where you analyze two numbers to see their relationship in terms of size and order. When dealing with inequalities, the main task is to see whether one side is greater than, less than, or equal to the other side.

• If the left hand side is smaller than the right side, the inequality \(<\) holds true.
• If it is larger or the same, it might be false or a different type of inequality will apply such as greater than \(>\) or equal \(=\).

In our problem, after performing the addition, the inequality \(22 < 23\) remains. Immediately, by glancing at 22 and 23, it is evident that 22 is less than 23. Thus, in this comparison step, we confirm that our inequality is true.

Addition

Addition is one of the basic mathematical operations you perform when solving inequalities or equations. It involves calculating the sum of two or more numbers. In the context of inequalities, addition helps in simplifying one side to compare with the other easily. For our exercise, after substituting the variable, the equation reads \(14 + 8 < 23\).

• The next logical step is to perform the addition of 14 and 8, which results in 22.
• This process reduces the inequality to a simpler form \(22 < 23\), which is easier to evaluate.

The key is to ensure the addition is performed correctly, preserving the inequality for accurate comparison later.

Problem-Solving

Problem-solving in mathematics involves systematically approaching a problem to find a solution. The approach often includes understanding the problem, devising a plan, carrying out the plan, and then reviewing the solution for accuracy. Applying this to inequalities, the following steps are essential:

• Understand the inequality you are working with.
• Substitute known values to simplify the inequality.
• Perform necessary operations like addition.
• Compare the results to see if the inequality holds true.

Practicing these steps prepares you to tackle a wide array of problems by breaking them down into manageable tasks.

True or False Statements

In mathematics, determining whether a statement is true or false is a basic skill essential for validating solutions. With inequalities, after performing operations like substitution and addition, you reach a point where you simply compare two numbers as done in our exercise.

• If the comparison holds, the statement is true.
• If it doesn’t, then it’s false. In our specific example, the statement \(22 < 23\) is proven quite evidently to be true, as 22 is indeed less than 23.

Understanding this concept reinforces the importance of logical reasoning and numerical accuracy in solving mathematical problems.