Problem 4 - Algebra for College Students

Question

In each of the following, translate part (a) as an expression and translate part (b) as an equation or inequality. Use \(x\) to represent the number. (a) 6 less than a number (b) 6 is less than a number.

Step-by-Step Solution

Answer

(a) \( x - 6 \); (b) \( 6 < x \)

Step 1 Title - Translate part (a)

To find an expression for '6 less than a number', let the number be represented by the variable \( x \). Therefore, '6 less than a number' can be written as \( x - 6 \). This completes the translation of part (a).

Step 2 Title - Translate part (b)

To find an equation or inequality for '6 is less than a number', again let the number be represented by the variable \( x \). The phrase '6 is less than a number' translates to the inequality \( 6 < x \). This completes the translation of part (b).

Key Concepts

expressions in algebra

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They allow us to represent real-world situations in a mathematical form. For example, in the exercise, the phrase '6 less than a number' is transformed into the expression \( x - 6 \). Here, 'a number' is represented by the variable \( x \), and '6 less than' means we are subtracting 6 from \( x \). It's crucial to understand how to form these expressions, as they are foundational in solving more complex algebraic problems.

• Expressions do not include equality or inequality signs.
• They can be simplified but not solved like equations.
• They often include a mix of coefficients (numeric values) and variables (unknown values).

Recognizing and writing algebraic expressions correctly helps in progressing to equations and inequalities.

inequalities in algebra

Inequalities in algebra describe the relative size or order of two values. Unlike equations that show equality, inequalities show how one value is less than, greater than, less than or equal to, or greater than or equal to another. In the exercise, '6 is less than a number' translates to the inequality \( 6 < x \). This tells us that the value of \( x \) must be greater than 6. It is the opposite direction of 'a number is less than 6,' which would be written as \( x < 6 \). Understanding inequalities is vital for graphing and solving real-world problems where exact values are not always required.

• Inequality symbols include: \(<, >, \leq, \geq \)
• They can be represented on number lines for visual understanding.
• Solutions to inequalities often include a range of values.

Practice with different phrases to become fluent in translating between words and inequalities.

variables in algebra

Variables are symbols used to represent unknown values or quantities in algebra. They are fundamental for constructing algebraic expressions, equations, and inequalities. In the exercise, the letter \( x \) is used as a variable to represent 'a number'. Using variables allows us to generalize mathematical problems and solve for unknowns in various contexts. When you see a variable, think of it as a placeholder for any number.

• Common variables include letters like \( x, y, z \).
• They can change value depending on the context of the problem.
• Using variables helps in forming general solutions that apply to many specific cases.

Grasping the role of variables aids in transitioning from simple arithmetic to more advanced algebraic concepts efficiently.