Problem 91 - Algebra A Combined Function

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. The sum of 5 and a number is 20 .

Step-by-Step Solution

Answer

The equation is \(5 + x = 20\).

Step 1: Identify the unknown number

In the given statement, 'a number' is our unknown. Let’s denote this unknown number by \( x \).

Step 2: Translate words to symbols

The phrase 'The sum of 5 and a number' translates to \( 5 + x \).

Step 3: Set up the equation

According to the statement, 'The sum of 5 and a number is 20', we can set this expression equal to 20, resulting in the equation: \( 5 + x = 20 \).

Key Concepts

Equations and Inequalities

Equations and inequalities are fundamental concepts in algebra that help us solve problems involving unknown values. An **equation** states that two expressions are equal. For instance, in our exercise, the equation is \( 5 + x = 20 \). It conveys a balance and tells you that whatever is on the left side is the same as what is on the right side. To find the value of \( x \), we solve the equation by manipulating the expressions while keeping this balance.

An **inequality**, on the other hand, indicates a relationship where two values are not necessarily equal, as they involve greater than, less than, or not equal to symbols, e.g., \( x > 5 \) or \( x \leq 10 \). While equations result in specific values for the unknowns, inequalities provide a range of possible solutions.

When learning these concepts, it is essential to pay attention to the language used and translate words into these mathematical relationships accurately. Practice will bolster your understanding and enable you to determine when to use an equation or inequality.

Translating Word Problems

Translating word problems into mathematical equations or inequalities is a critical skill. It involves reading a sentence or problem and expressing its meaning using symbols and numbers. For example, in the sentence "The sum of 5 and a number is 20," we must identify keywords and numbers.

• "Sum" suggests addition.
• "A number" refers to our unknown, which we denote by \( x \).
• "Is" typically represents the equality sign \( = \).

When all these elements come together, the phrase turns into the equation \( 5 + x = 20 \).

Practicing this method will help strengthen your ability to represent different types of word problems with mathematical expressions. The more you practice, the better you'll become at identifying the meaning behind each word and constructing clear equations and inequalities.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In our exercise, the phrase "the sum of 5 and a number" is represented as the algebraic expression \( 5 + x \). These expressions form the building blocks of equations and can represent situations or problems explicitly when translated correctly from a word problem.

Variables in the expressions, such as \( x \) in our example, signify numbers that can change or are unknown. The operations, such as addition, subtraction, multiplication, or division, define the relationship between these variables and constants (known numbers).

• Understand that a variable can be any number or range of numbers, depending on the context.
• Expressions simplify problems by symbolizing real-world situations.
• Learn to manipulate expressions to isolate a variable or simplify parts of a problem.

Practicing writing and simplifying algebraic expressions will enhance your problem-solving skills and deepen your understanding of algebra.