Problem 47 - Algebra 2

Question

Define a variable and write an inequality for each problem. Then solve. Twice the sum of a number and 5 is no more than 3 times that same number increased by 11.

Step-by-Step Solution

Answer

The inequality is \( x \geq -1 \).

Step 1: Define the Variable

Let the unknown number be represented by the variable \( x \). This variable will be used throughout the inequality.

Step 2: Formulate the Inequality

The problem states that twice the sum of a number and 5 is no more than 3 times the same number increased by 11. This can be written as: \[ 2(x + 5) \leq 3x + 11 \]

Step 3: Expand and Simplify the Inequality

First, expand the left side of the inequality: \[ 2(x + 5) = 2x + 10 \] So the inequality becomes: \[ 2x + 10 \leq 3x + 11 \]

Step 4: Rearrange the Inequality

Subtract \( 2x \) from both sides to get: \[ 10 \leq x + 11 \]

Step 5: Isolate the Variable

Subtract 11 from both sides to solve for \( x \): \[ 10 - 11 \leq x \] Which simplifies to: \[ -1 \leq x \]

Step 6: Write the Solution in Set Notation

This inequality can be expressed as: \[ x \geq -1 \] It indicates that the number must be greater than or equal to -1.

Key Concepts

Understanding Variables

A variable is a symbol, commonly a letter, used to represent an unknown number or value in mathematical expressions or equations. In this exercise, we use the variable \( x \) to stand for the unknown number mentioned in the problem. The choice of letter doesn’t affect the problem-solving process, though \( x \) is often used for convenience.

Variables allow us to write equations and inequalities in a general form, enabling the representation of a wide range of possible numbers. It’s much like a "placeholder" in an equation that can take on different values. Without variables, describing relationships between numbers would be cumbersome and less efficient.

When working with variables, remember that the goal is usually to solve for this unknown value. This involves manipulating algebraic expressions to isolate the variable and determine its value based on the given conditions.

Steps in Inequality Solving

Solving inequalities is similar to solving equations, with a few critical differences. When working with inequalities, one manipulates expressions to isolate the variable just like in equations, but must also consider the direction of the inequality sign.

• **Step 1:** Simplify and expand both sides if necessary. This helps in understanding the relationship between the terms.
• **Step 2:** Rearrange or combine like terms to move them across the inequality sign.
• **Step 3:** Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality.
• **Step 4:** Pay attention to dividing or multiplying by negative numbers, as it reverses the inequality sign.

In our example, simplifying the sides and rearranging leads us to the inequality \( x \geq -1 \), indicating the values that satisfy the given conditions.

Exploring Inequality Notation

Inequality notation is a mathematical language that describes the relative size or order of two values or expressions. It doesn’t state that two things are equal, but rather defines a range of possible values.

• The symbol \( \leq \) means "less than or equal to".
• The symbol \( \geq \) means "greater than or equal to".
• The symbol \( < \) means "less than".
• The symbol \( > \) means "greater than".

In our problem, we arrive at the inequality \( x \geq -1 \), which means \( x \) can be any number that is -1 or greater. Using inequality notation, we can easily communicate and understand the solution set in mathematical and real-world contexts.

Working with Algebraic Expressions

Algebraic expressions are an essential part of mathematics, combining numbers and variables with operations like addition, subtraction, multiplication, and division. In the given problem, we encounter the algebraic expression \( 2(x + 5) \).

Understanding and manipulating such expressions are key skills in algebra. To solve for \( x \), we expanded \( 2(x + 5) \) into \( 2x + 10 \). This process of expanding helps to simplify expressions or prepare them for solving equations or inequalities.

Another important aspect is combining like terms, which involves grouping and simplifying expressions by their similar components. This step-by-step manipulation of expressions uncovers the relationships between terms and helps to solve the given inequality. An organized approach makes solving them much more straightforward.