Problem 66 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. Let x represent the number. 5 less than a number is \(8 .\)

Step-by-Step Solution

Answer

The sentence '5 less than a number is 8' translates into the equation \(x - 5 = 8\).

Step 1: Identify the Number

The phrase 'a number' is referred as 'x'. So, our unknown number will be represented by the letter 'x'.

Step 2: Translate '5 less than a number'

The phrase '5 less than' can be considered as subtraction. So, '5 less than a number' translates into 'x - 5'. It's important to note that the phrase '5 less than a number' means 5 has to be subtracted from the number, not the other way round.

Step 3: Translate 'is'

The verb 'is' can be translated into '='. So, '5 less than a number is 8' translates into 'x - 5 = 8'.

Key Concepts

Equations

Equations are mathematical statements that assert the equality of two expressions.
The phrase "5 less than a number is 8" exemplifies an equation in verbal form. By identifying the unknown number as "x," we can represent this scenario mathematically.
When solving equations, our goal is to find the value of this unknown number, or "x," that makes the equation true. In the given problem, "5 less than a number is 8" translates into the equation:
\[ x - 5 = 8 \]
The equal sign "=" is a key feature distinguishing equations from other mathematical statements, such as inequalities.
To solve an equation like \( x - 5 = 8 \), you need to isolate "x" by performing inverse operations.

• Add 5 to both sides of the equation to cancel out the subtraction of 5.
• This leads to the equation \( x = 13 \), showing the number that satisfies the original condition.

Understanding equations is essential in algebra as they form the foundation for expressing and solving problems involving unknown quantities.

Inequalities

Inequalities, unlike equations, are mathematical statements that compare the relative size or order of two values. They do not claim equality but show that one side is less than, greater than, or not equal to the other.
Some common inequality symbols include:

• \(<\) : less than
• \(>\) : greater than
• \(\leq\) : less than or equal to
• \(\geq\) : greater than or equal to

In the context of the problem where "5 less than a number is 8," the situation naturally forms an equation because equality is established. However, inequalities are useful when there are conditions such as ranges or limits instead of precise values.
For example, if the statement was "5 less than a number is less than 8," it would be expressed as:
\[ x - 5 < 8 \]
To solve inequalities, similar operations as in equations are applied with extra care around multiplying or dividing variables by negative numbers, which can reverse the inequality sign.
Inequalities allow us to represent situations where there is more flexibility in numbers, making them a powerful tool in algebra and beyond.

Translation of Phrases

Translating verbal phrases into mathematical expressions is a crucial skill in algebra. Breaking down expressions such as "5 less than a number is 8" into symbols involves identifying key terms and their operations.
Here's a simple guide for translating phrases:

• Identify the unknown number with a variable, often "x."
• Recognize and represent operations: "less than" implies subtraction, "more than" implies addition, "times" implies multiplication, and "divided by" implies division.
• Translate relational terms: "is" becomes equals "=", while "less than", "greater than" lead to inequalities.

For the exercise, the phrase "5 less than a number" converts to "x - 5" because it indicates a reduction from the number by 5.
The key is to maintain the order of the phrases to accurately reflect their mathematical meaning.
Practicing this translation helps sharpen problem-solving skills and better equips students to tackle complex algebraic expressions.
Once you master translation, working with equations and inequalities becomes much more intuitive.