Problem 9 - Algebra and Trigonometry

Question

\(9-16\) State whether each inequality is true or false. $$ -6 < -10 $$

Step-by-Step Solution

Answer

The inequality -6 < -10 is false.

Step 1: Understand the Symbols

In the inequality \(-6 < -10\), the symbol \<\(<\) means "is less than." This implies that we want to see if -6 is a smaller number than -10.

Step 2: Compare the Numbers

Look at the values -6 and -10 on the number line. The number that is further to the right is greater. Since -6 is to the right of -10 on the number line, this means that -6 is actually greater than -10.

Step 3: Determine the Truth of the Inequality

Since -6 is greater than -10, the statement \(-6 < -10\) is false. The inequality sign is incorrect according to the actual positions of the numbers on the number line.

Key Concepts

Number Line Comparison

A number line is a visual tool that helps when comparing the size of numbers. It is a straight line with numbers placed at equal intervals along its length. When using a number line for comparison, remember these key points:

• Numbers increase in value as you move to the right.
• Negative numbers are situated to the left of zero.
• The further right a number is positioned, the larger it is.

For example, when comparing \(-6\) and \(-10\), imagine where these numbers sit on the number line. \(-6\) is further to the right than \(-10\), indicating that \(-6\) is larger. This visualization helps to decide whether an inequality like \(-6 < -10\) holds true or false.
Using a number line consistently in math can make it easier to understand and visualize the relationships between different values, especially with negative numbers.

Inequality Symbols

Inequalities are a way of showing how quantities compare to each other. The symbols used in inequalities are crucial to understanding what the statements mean:

• \(>\) means "greater than."
• \(<\) means "less than."
• \(\geq\) means "greater than or equal to."
• \(\leq\) means "less than or equal to."

In this exercise, the symbol used is \(<\), which asks if one number is less than another. It is also important to note that these symbols are directional on a number line. With \(-6 < -10\), we look to see if \(-6\) lies to the left of \(-10\) on the number line. Understanding which inequality symbol to use accurately reflects the relationship between numbers.

Truth Value of Inequalities

The truth value of an inequality indicates whether the statement is correct. Determining this involves ensuring the direction of the inequality matches the actual relationship of the numbers involved. For the inequality \(-6 < -10\), the expected comparison is that \(-6\) should be smaller than \(-10\). However, on a number line, \(-6\) is to the right of \(-10\), which actually makes it larger.

• If the statement matches the reality of number placement, it is true.
• If not, it is false.

In this problem, because \(-6\) is not less than \(-10\), the inequality \(-6 < -10\) is false. Checking the truth value is essential to ensure the mathematical statement reflects the apparent numerical relationship.