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Chapter 1: Problem 62

Classify each inequality as either true or false. $$8 \geq 8$$

Short Answer

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Step by step solution

01

Identify the Symbols

The symbols used in the inequality are important. The symbol \(\textgreater \text{or equal to}\rightarrow\geq\) means that the left-hand side (8) should either be greater than or equal to the right-hand side (8) for the inequality to be true.
02

Compare the Left-Hand and Right-Hand Side

Next, rewrite the inequality in basic terms: 8 is greater than or equal to 8. We need to assess whether 8 is either greater than or equal to 8.
03

Evaluate the Inequality

Compare the two values. Since 8 is equal to 8, the condition of 'greater than or equal to' is met. Thus, the inequality \(\textgreater \text{or equal to}\rightarrow\geq\) is true in this case.

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Key Concepts

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

Greater than or equal to
The symbol \(\geq\) is a fundamental part of inequalities in mathematics. It reads as 'greater than or equal to'. This means that when you see \(a \geq b\), the value on the left-hand side (a) can either be more than or exactly equal to the value on the right-hand side (b).

For example, consider \(8 \geq 8\). Here, we need to check if 8 is greater than or equal to 8. Since 8 is equal to 8, the inequality is true.

This symbol is very useful in cases where we include the boundary value in a comparison. So, anytime you see \(\geq\), remember it includes the equality as well as the greater-than condition.
Evaluating inequalities
Evaluating inequalities involves determining whether the relationship expressed by the inequality is true. To do this, you follow a systematic approach:
  • First, identify the inequality symbol and understand its meaning.
  • Second, compare the values on both sides of the inequality.
  • Finally, decide if the statement is true based on the comparison.

For the inequality \(8 \geq 8\), let's break it down:

  • The symbol \(\geq\) means 'greater than or equal to'.
  • We compare the two values: 8 and 8.
  • Since 8 is equal to 8, the inequality is true.

Following these steps systematically helps to accurately determine the truth value of any inequality.
Algebraic comparison
Algebraic comparison is the process of comparing algebraic expressions or numbers to determine their relationship. It forms the basis for solving and understanding inequalities. Here are key points to remember:
  • Always pay attention to the inequality symbol since it dictates the kind of comparison.
  • Compare the corresponding sides of the inequality to evaluate it.
  • Use number properties and basic math operations for the comparison.

In the given example, \(8 \geq 8\), there are no variables or complex expressions. We directly compare the two numbers.

Since 8 equals 8, our comparison shows that the inequality holds true.

This method can also be used for more complex inequalities that involve variables and multiple steps. Remember, clear comparison is the key to solving inequalities correctly.

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Most popular questions from this chapter

Translate to an inequality. When Helga's calculator gives a decimal value for \(\sqrt{2}\) and that value is promptly squared, the result is \(2 .\) Yet when that same decimal approximation is entered by hand and then squared, the result is not exactly \(2 .\) Why do you suppose this is?

Write an equivalent expression without using grouping symbols. $$ -[5 x-2 y-3 z] $$

Simplify. $$ 15 x-y-5(3 x-2 y+5 z) $$

SYNTHESIS Is it true that for all real numbers \(a\) and \(b\) $$ a b=(-a)(-b) ? $$ Why or why not?

Evaluate. $$ -30 \div t(t+4)^{2}, \text { for } t=-6 $$
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