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Chapter 5: Problem 42

Determine whether each inequality is true or false. $$31-17<12$$

Short Answer

Expert verified
False; 31-17 is 14, which is not less than 12.

Step by step solution

01

Calculate the Left Side

To solve the inequality, we first need to calculate the left side. The left side of the inequality is given as \(31 - 17\). We perform the subtraction: \(31 - 17 = 14\).
02

Compare Left to Right

Now that we have calculated the left side as 14, we compare it with the right side of the inequality, which is 12. This comparison is needed to determine if the inequality \(14 < 12\) is true or false.
03

Determine Inequality Truth

Since 14 is not less than 12, the inequality \(14 < 12\) is false. Therefore, the original inequality \(31 - 17 < 12\) is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Mathematical Reasoning
Mathematical reasoning is an important skill in solving inequalities. It involves logical thinking and the application of mathematical principles to determine relationships between quantities.
When you're faced with an inequality like the one here, 31 - 17 < 12, you need to apply reasoning to each step:
  • First, understand the goal. Here, it's to compare the difference between two numbers to a third number.
  • Next, perform operations methodically. This involves calculating each side accurately.
  • Finally, assess the truthfulness of the statement. Compare results to see if the inequality holds.
Each of these steps requires reasoning to ensure correct and meaningful results. Even a simple mistake in thinking can lead to errors in your final conclusion, making mathematical reasoning vital.
Mastering Comparison Operations in Inequalities
Comparison operations are building blocks in mathematics, especially when dealing with inequalities. These operations help compare two quantities to see which one is larger, smaller, or if they're equal.
In the inequality 31 - 17 < 12, the operation involved is the "less than" operation, symbolized as '<'.
Here are some comparison symbols and their meanings:
  • <: Less than.
  • >: Greater than.
  • =: Equal to.
  • 鈮�: Less than or equal to.
  • 鈮�: Greater than or equal to.
Understanding how to interpret these symbols is essential when analyzing mathematical statements. It allows you to determine relations between numbers and decide the validity of inequalities effectively.
Applying Subtraction in Inequalities
Subtraction is a fundamental arithmetic operation and plays a crucial role in solving inequalities. In our problem, we need to evaluate 31 - 17 to determine how it compares to 12.
Here's a simple guide to using subtraction within inequalities:
  • Always work on one side of the inequality first. Here, solve 31 - 17 to get a result, which is 14.
  • Next, compare this result with the other side of the inequality. For our example, compare 14 with 12.
  • Determine if the inequality holds true based on your result.
By mastering subtraction and correctly evaluating each side, you can accurately determine whether an inequality like the one given is true or false. Always double-check your subtraction for accuracy to avoid mistakes.

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