Problem 25 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. Let x represent the number. A number increased by 10 is greater than or equal to 44.

Step-by-Step Solution

Answer

The equation representing the sentence is \(x + 10 ≥ 44\)

Step 1: Represent the Unknown

Here 'a number' is mentioned in the problem. Let 'x' represent this unknown number.

Step 2: Translate the First Part

'A number increased by 10' translates to 'x + 10' in the equation. The word 'increased' indicates addition.

Step 3: Translate the Inequality

'is greater than or equal to' translates to '≥' in the equation.

Step 4: Write the Final Number

The number '44' stays the same in our equation and is written as such.

Step 5: Write the Complete Equation or Inequality

Combine the translations from Steps 2, 3, and 4 to form a complete equation or inequality: 'x + 10 ≥ 44'

Key Concepts

Representing Unknowns

In math, an unknown typically refers to a number we don't have a specific value for just yet. We use a variable, often a letter like \( x \), to stand in for this unknown number. This approach makes it easier to work with equations and to solve problems. Think of \( x \) as your placeholder, indicating that there is a value, but it isn't defined yet.
Imagine you're solving a puzzle, and \( x \) is the missing piece. It's the number that, once discovered, will complete the equation.

• Unknowns are common in various math problems.
• We often denote unknowns with letters like \( x \), \( y \), or \( z \).
• Using variables lets us write flexible and universal algebraic expressions.

This foundational concept paves the way for understanding more complex mathematical problems.

Translating Verbal Expressions

Math problems often start with verbal expressions or statements. The key is to translate these words into mathematical terms or actions. For example, when we read "a number increased by 10," we need to identify the operation happening here. The word "increased" tells us we are adding. Thus, you can express "a number increased by 10" with a mathematical expression: \( x + 10 \).
Translating word problems into mathematical expressions is crucial because it allows us to apply arithmetic and algebraic operations to find solutions.

• Identify key phrases and words, such as "sum," "difference," "product," and their algebraic meanings.
• Translate them into mathematical symbols, like \(+\), \(-\), \(\times\), or \(\div\).
• The goal is to form a concise and accurate mathematical expression from the verbal statement.

Once you become comfortable with this translation process, word problems will become much more manageable.

Writing Equations and Inequalities

Equations and inequalities are essential tools in mathematics, used to express relationships and restrictions. In our example, the keywords "is greater than or equal to" translate into the inequality symbol \( \geq \). Therefore, the translated verbal statement becomes the inequality \( x + 10 \geq 44 \).
An equation shows two expressions are equal, while an inequality indicates that they have a certain inequality relationship.

• "=": The symbol for equations, meaning both sides are equal.
• "\(<\)" and "\(>\)": Symbols indicating less than or greater than respectively, used for inequalities.
• "\(\leq\)" and "\(\geq\)": Symbols for less than or equal to, and greater than or equal to, showing more specific relationships.

Understanding how to write these expressions accurately allows you to set up and solve a wide variety of mathematical problems effectively.