Problem 1 - 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) 15 more than a number (b) 15 is more than a number.

Step-by-Step Solution

Answer

(a) \( x + 15 \); (b) \( 15 > x \)

Step 1: Translate Part (a)

Identify the key phrase '15 more than a number'. This means adding 15 to a number represented by x. So the expression is: \[ x + 15 \]

Step 2: Translate Part (b)

Identify the key phrase '15 is more than a number'. This suggests an inequality where 15 is greater than a number represented by x. So the inequality is: \[ 15 > x \]

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in equations and expressions. The symbols often stand for unknown values or variables. In our exercise, we are using the variable \( x \) to represent an unknown number.

• Variables help in forming general rules to solve various mathematical problems.
• Expressions are combinations of variables, numbers, and operators (like + or -).

The expression \( x + 15 \) shows us how algebra can be used to translate a phrase like '15 more than a number' into mathematical terms. By mastering algebra, you will understand how to simplify and solve these problems easily.

Translating and Understanding Inequalities

Inequalities are mathematical sentences that compare two values or expressions using inequality symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). In the given problem, the inequality \[ 15 > x \] can be interpreted by reading the phrase '15 is more than a number' which means 15 is greater than the unknown number represented by \( x \).

• Understanding inequalities helps us in determining ranges or possible values for variables.
• Inequalities are crucial in real-life situations like comparing trends or finding limits.

Practice converting phrases to inequalities; it makes recognizing the relationships much easier.

Working with Expressions

An expression in algebra is a combination of one or more numbers, variables, and arithmetic operations. Operations could include addition, subtraction, multiplication, or division. In our example, the expression \( x + 15 \) takes the idea of '15 more than a number' and converts it into a mathematical form you can work with.

• Expressions do not have an equality sign (unlike equations); they show a relation between variables and constants.
• Expressions can be simplified or evaluated for specific values of variables.

The step-by-step approach you see here – breaking down a problem, identifying keywords, and translating them – is very useful not only in algebra but in solving a wide variety of mathematical problems. Practice with various phrases to become comfortable with creating and simplifying expressions.