Problem 43 - Algebra and Trigonometry

Question

True or false. $$-13 \leq-2$$

Step-by-Step Solution

Answer

True

Step 1: Interpret the Inequality Sign

The inequality \(\leq\) means 'less than or equal to'. So \(-13 \leq -2\) reads as '-13 is less than or equal to -2'.

Step 2: Understanding Negative Numbers

On the number line, numbers to the left are smaller than those to the right. Negative numbers are to the left of 0, and -13 is to the left of -2. So -13 is less than -2.

Step 3: Validating the Inequality

Since -13 is indeed less than -2, the inequality \(-13 \leq -2\) is true.

Key Concepts

Negative Numbers

Understanding negative numbers is a fundamental concept in mathematics that often appears in topics like inequalities. Negative numbers are those numbers that are less than zero. They are represented with a minus sign in front, such as

• -1
• -13
• -100

Negative numbers can be thought of as points on the left side of zero on a number line. The further left a number is from zero, the smaller it is considered to be. For instance,

• -13 is smaller than -2
• -50 is smaller than -10

This is because -13 is more to the left on the number line compared to -2. Understanding the relation between negative numbers helps in determining their size relative to each other, which is crucial when solving inequalities or interpreting mathematical statements like the one in this exercise.

Number Line

The number line is a visual and straightforward way to understand the ordering of numbers, including both positive and negative values. It is a straight line where numbers are placed at equal intervals. Zero is the central point, negative numbers stretch to the left, and positive numbers extend to the right.
When examining inequalities, a number line can help you see which numbers are larger or smaller. In the exercise example, -13 lies to the left of -2 on the number line.
This means

• -13 is less than -2

Thus, the inequality

• -13 \[\leq\] -2

illustrates that -13 is equivalent to being lesser or equal, which confirms the statement is true. Utilizing the number line as a tool in understanding mathematical concepts provides clarity and enhances comprehension, particularly when dealing with inequalities and comparisons of negative numbers.

Step by Step Solution

Solving problems using a step-by-step approach can simplify even the most complex mathematical concepts. Let's break down the process using the given exercise:

• **Step 1: Interpret the Inequality Sign** - The inequality \[\leq\] symbolizes 'less than or equal to', so when you see \[-13 \leq -2\] it reads as '-13 is less than or equal to -2.'
• **Step 2: Understanding Negative Numbers** - On a number line, numbers to the left are less than those on the right. Knowing that -13 is further left than -2 helps to easily see that -13 is indeed less than -2.
• **Step 3: Validating the Inequality** - The final step is confirming the inequality. With the knowledge that -13 is smaller (-13 \[\leq\] -2), the statement is validated as true.

By dissecting problems into smaller, manageable steps, students can methodically work through each part to reach an accurate solution. This step-by-step tactic not only aids in solving inequalities but also improves overall problem-solving skills.