Question

Which of these expressions states that \(x\) is less than or equal to 3 ? (A) \(x<3 \quad\) OR (B) \(\quad x \leq 3\)

Short answer

The correct expression is (B) \(x \leq 3\).

Step-by-step solution

Understanding the question

We must determine which of the given expressions represents the scenario where the value of \(x\) is less than or equal to 3.

Analyzing option (A)

The expression \(x<3\) states that \(x\) is strictly less than 3, meaning \(x\) can be any value that is smaller than 3, but not equal to 3.

Analyzing option (B)

The expression \(x \leq 3\) means \(x\) is less than or equal to 3, covering all values less than 3 as well as the value equal to 3.

Comparing both options

Option (A) excludes the possibility of \(x\) being equal to 3, while option (B) includes it. The expression "less than or equal to" is specifically noted in option (B).

Key concepts

Understanding Algebra in Inequalities

Algebra serves as a foundational pillar in mathematics, where we use symbols and letters to represent values and solve problems. When dealing with inequalities, algebra helps us precisely express relationships between different mathematical expressions. In an inequality, such as \(x \leq 3\), algebra explains that \(x\) can be any value up to and including 3.

Inequalities differ from equations. In equations like \(x = 3\), \(x\) has a single value. Inequalities, on the other hand, offer a range of possibilities. This is because inequalities use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to show a comparison.

• \(<\): Less than
• \(>\): Greater than
• \(\leq\): Less than or equal to
• \(\geq\): Greater than or equal to

Inequalities are crucial in understanding how variables interact and the possible values they can hold.

Breaking Down Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators, such as addition and subtraction. In the context of inequalities, expressions like \(x < 3\) and \(x \leq 3\) precisely articulate the limits of \(x\).'

Each part of these expressions has a role. For example, in \(x \leq 3\), the "\(x\)" represents the variable. The "\(\leq\)" symbol communicates the relationship of the variable concerning the number "3." This signifies all values \(x\) could be, indicating it could be any number up to and including 3.

• The variable (e.g., \(x\)) represents values that can change.
• Operators (e.g., \(<\), \(\leq\)) define the relationship between variables and numbers.
• Numbers (e.g., 3) set limits or specific values in the expression.

Understanding expressions is essential for solving algebraic problems correctly and efficiently.

Solving Problems with Inequalities

Problem-solving with inequalities involves determining the possible values of a variable that satisfy the given condition. In this exercise, you must decide between two expressions: \(x < 3\) and \(x \leq 3\). Choosing correctly requires understanding what the inequality symbols represent.

Option \(A\), \(x < 3\), limits \(x\) to values strictly less than 3, excluding 3 itself. Option \(B\), \(x \leq 3\), allows \(x\) to be any value less than or equal to 3. Identifying this distinction is crucial for accurately interpreting and solving inequality problems.

• Carefully read each symbol's meaning.
• Visualize the solution set on a number line to see the range of possible values.

Grasping these concepts enhances problem-solving skills and enables you to tackle algebraic challenges with confidence.