Problem 12 - Algebra 1

Question

Write the verbal sentence as an equation or an inequality. Ten more than a number \(x\) is greater than fourteen.

Step-by-Step Solution

Answer

The inequality to represent the verbal sentence is \(x + 10 > 14\).

Step 1: Identify the Unknown

The sentence mentions 'a number', which is not specifically identified. This is the variable or the unknown in the sentence. Let's represent the unknown number as \(x\).

Step 2: Translate 'Ten More Than A Number'

'Ten more than a number' means that you add 10 to the number. In algebraic expressions, this is written as \(x + 10\).

Step 3: Translate 'Is Greater Than'

'Is greater than' is a comparative phrase that shows a larger quantity. In mathematical notation, 'is greater than' is represented by the symbol '>'. The expression grows into \(x + 10 > \ldots\)

Step 4: Translate 'Fourteen'

This is the value being compared to. In the initial sentence, it states that 'ten more than a number is greater than fourteen'. It implies \(x + 10 > 14\).

Key Concepts

Algebraic Expressions

When we talk about algebraic expressions, we're referring to a combination of numbers, variables, and mathematical operators (like addition or multiplication) that represent a specific value. Imagine you're working with building blocks, where each block represents a number or a variable, and you're joining these blocks according to given rules or operations.

For instance, if I say 'three more than a number', in algebra, we'd write that as 'number + 3'. This is now an algebraic expression because it contains a variable (the 'number') and an operation (in this case, addition). Remember, algebra is like a universal language — it lets us translate words into symbols that mathematicians all over the world can understand!

Inequalities

Inequalities are statements that compare two expressions and determine if one is larger, smaller, or not equal to the other. They're like the scales of justice, balancing or tipping to show which side is heavier (greater) or lighter (lesser).

In shapes, sizes, and even quantities, not everything is created equal, and that's what inequalities help us express. Mathematical symbols for inequalities include '>' (greater than), '<' (less than), '≥' (greater than or equal to), and '≤' (less than or equal to). These symbols are the heartbeat of our inequality statements, giving us the pulse on which side tips the scale.

Variable Representation

Variables are symbols used to represent numbers in equations and expressions, typically letters like 'x', 'y', or 'z'. Think of a variable as a placeholder or a mystery box which could be filled with any number that fits the rules of the equation or expression.

In our original problem, we chose the letter 'x' to stand for 'a number'. This is variable representation at its core: a simple and effective way to highlight that while we don't know the exact value yet, we can still work with it, analyze it, and solve problems involving it. Whether in algebra class or a scientist's lab, variables are essential components of the problem-solving toolkit.