Problem 6 - Intermediate Algebra

Question

Fill in the blanks. To _____ an inequality means to find all values of the variable that make the inequality true.

Step-by-Step Solution

Answer

The word is 'solve'.

Step 1: Identify the Task

The task is to find the missing word that describes the action of determining all values of the variable that satisfy the inequality.

Step 2: Recall Mathematical Vocabulary

In mathematics, the term used to describe the process of finding all values that satisfy an inequality is similar to determining the solution of an equation.

Step 3: Apply Vocabulary Knowledge

The word 'solve' is commonly used to describe finding all solutions that satisfy an inequality or an equation.

Key Concepts

Solving Inequalities

When we talk about solving inequalities, we are focusing on finding all the possible values for a variable that make the inequality statement true. An inequality might look similar to an equation but instead of an equal sign, it employs symbols like <, >, \( \leq \), or \( \geq \). These symbols indicate whether one side is less than, greater than, less than or equal to, or greater than or equal to the other side.

Here's a brief overview of how to solve inequalities:

• Identify the inequality and the variable whose values you need to find.
• Use addition, subtraction, multiplication, or division to isolate the variable on one side of the inequality.
• Remember that if you multiply or divide the inequality by a negative number, the inequality symbol must be flipped. For example, if you have \( -2x > 4 \), dividing both sides by -2 gives \( x < -2 \).
• Once the variable is isolated, the resulting statement will show the range of values that make the inequality true.

By following these steps, you can solve inequalities methodically, just as you would solve equations.

Mathematical Vocabulary

Mathematical vocabulary forms the backbone of solving mathematical problems. Understanding the right terms can significantly ease the process of solving problems like inequalities.

Some important terms you should be familiar with include:

• Solve: To find all the values that satisfy the given condition or equation.
• Solution: The set of all possible values of the variable that make the inequality true.
• Inequality Signs: Symbols such as <, >, \( \leq \), and \( \geq \) used to express the relationship between expressions.
• Isolate: To get a variable alone on one side of an equation or inequality.

Familiarity with these terms will help you not only in solving inequalities but also in advancing to more complex mathematical concepts.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols to solve problems. In the context of inequalities, algebra helps us understand how to manipulate and solve inequalities involving variables.

The basic building blocks of algebra include:

• Variables: Symbols (often letters) used to represent numbers in equations and inequalities.
• Expressions: Combinations of variables, numbers, and operations (like addition and multiplication).
• Equations: Mathematical statements that assert the equality of two expressions.
• Inequalities: Statements that compare expressions to show that one is greater or less than the other.

Algebraic skills are essential in solving inequalities because they allow you to manipulate and transform expressions to find the solution. Mastering these concepts will prepare you for solving a wide range of mathematical problems.