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Chapter 0: Problem 9

The statement \(x

Short Answer

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Step by step solution

01

Identify the Meaning of the Inequality

The statement provided is an inequality: \(x < y\). This means that the value of \(x\) is smaller than the value of \(y\).
02

Visualize on a Number Line

Consider the number line. Smaller values lie to the left, and larger values lie to the right. Therefore, if \(x\) is smaller than \(y\), \(x\) must lie to the left of \(y\).
03

Conclude the Position

Given \(x < y\), conclude that \(x\) lies to the left of \(y\) on the number line.

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Key Concepts

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

Number Line
Understanding the number line is crucial for visualizing inequalities in algebra. A number line is a horizontal line where each point corresponds to a real number.

On a number line:
  • Smaller values are placed to the left.
  • Larger values are placed to the right.

If we consider the inequality statement ewline--- \(x < y\) --- ewline this tells us that the value of \(x\) is smaller than \(y\). Thus, on a number line, \(x\) would be positioned to the left of \(y\).

This visual representation makes it easier to understand and solve inequalities in algebra.
Inequality Notation
Inequality notation is a way to express the relative size or order of two values. This is different from equality, which shows that two values are the same.

The common symbols used in inequality notation are:
  • \('<'\): Less than
  • \('>'\): Greater than
  • \('\leq'\): Less than or equal to
  • \('\geq'\): Greater than or equal to

For example, \(x < y\) denotes that \(x\) is less than \(y\), meaning \(x\) is positioned to the left of \(y\) on the number line. Similarly, \(x > y\) would mean \(x\) is to the right of \(y\).

Understanding these symbols and their meanings helps in graphing and solving algebraic equations involving inequalities.
Algebra Basics
Before diving deep into inequalities and their visualizations, it's important to have a solid understanding of basic algebra concepts.

At its core, algebra involves:
  • Variables: Symbols that represent unknown values.
  • Expressions: Combinations of variables and constants (like numbers) using operations (like addition and multiplication).
  • Equations: Statements that two expressions are equal, often solved to find the value of variables.
Understanding these basics is essential for tackling more advanced topics, including inequalities.

With the foundational knowledge of algebra, you will find it easier to grasp more complex concepts, such as representing and solving inequalities on a number line.

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Most popular questions from this chapter

Multiply and simplify. $$ (w+7)(6 w-3) $$

Use a calculator to approximate the expression to 2 decimal places. $$\frac{-3+5 \sqrt{2}}{7}$$

Use a calculator to approximate the expression to 2 decimal places. $$\frac{6-3 \sqrt{5}}{4}$$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 4 \sqrt{10}(7 \sqrt{2}-3 \sqrt{5}+\sqrt{10}) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (2 \sqrt{3}+\sqrt{7})(2 \sqrt{3}-\sqrt{7}) $$
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