Lesson 45: Translating Between Words and Inequalities

New Concept

Algebra is a mathematical language that uses symbols to represent unknown quantities and relationships, like equations and inequalities, to solve real-world problems.

What’s next

This is the foundational skill of algebra. Next, you’ll practice this translation with a specific tool: inequalities, working through examples and real-world applications.

Property

An inequality is a mathematical statement comparing quantities that are not equal. The symbols used are $<$ (less than), $>$ (greater than), $≤$ (less than or equal to), $≥$ (greater than or equal to), and $≠$ (does not equal).

Examples

• x > 5 means a number is greater than 5.
• c + 2 < 10 means the sum of a number and 2 is less than or equal to 10.
• p ≠ 0 means a number is not equal to 0.

Explanation

Think of inequalities as see-saws that aren't perfectly level! They show one side is heavier, lighter, or perhaps equal, but never just the same. They help us compare values that have a range of possibilities instead of a single fixed answer, which makes math much more flexible and fun.

Property

To translate a sentence into an inequality, identify the key operations (sum, product, quotient) and inequality phrases ('is less than', 'is at least') to construct the correct mathematical expression.

Examples

• "The quotient of a number and 2 is less than or equal to 6" translates to $\frac{n}{2} \leq 6$.
• "The sum of the product of 20 and a number and 75 is at least 195" translates to $20x + 75 \geq 195$.
• "The difference of a number and 2.8 does not equal 8.2" translates to $g - 2.8 \neq 8.2$.

Explanation

This is like being a codebreaker! You're translating English into the language of math. Look for clue words like 'quotient' for division or 'at least' for $≥$. Each word gives you a piece of the puzzle, letting you build the final inequality that solves the mystery.

Property

To write an inequality as a sentence, replace the mathematical symbols for operations (+, -, multiplication, division) and comparisons (<, ≥) with their corresponding English words or phrases.

Examples

• x + 8 > 6 can be read as "The sum of a number and 8 is greater than 6."
• 2.5z < 15 can be read as "The product of 2.5 and a number is less than 15."
• 3x - 6 ≤ -30 can be read as "The difference of 3 times a number and 6 is less than or equal to -30."

Explanation

Now you're the storyteller! Turn a math sentence like $x + 8 > 6$ into a real sentence. You just need to know the 'language' of the symbols, like knowing + means 'the sum of' and > means 'is greater than'. It’s a reverse translation challenge!

Property

To solve real-world problems with inequalities, determine the unknown and assign it a variable. Then, translate phrases describing limits, like 'at most' ($≤$) or 'at least' ($≥$), into a mathematical inequality.

Examples

• A team has at most 1000 dollars for 15 suits and a 25 dollars fee. This becomes $15c + 25 \le 1000$.
• A skater with 45.7 points needs a score of 83.2 or better to win. This becomes $45.7 + s \ge 83.2$.

Explanation

Inequalities are your superpower for making smart decisions with limits, like a budget! Figure out the maximum you can spend or the minimum you need to achieve. It’s all about using math to set boundaries and find all the possible winning answers that exist within those rules.