Problem 31 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ -8

Step-by-Step Solution

Answer

\(-15 < x\) or {x | x > -15}

Step 1: Understand the Inequality

We start with the inequality \(-8 < x + 7\). Our goal is to solve for \(x\) and express the solution in solution set notation.

Step 2: Isolate the Variable

To isolate \(x\), we need to eliminate the constant on the right side. Subtract 7 from both sides of the inequality:\[-8 - 7 < x + 7 - 7\] which simplifies to:\[-15 < x\]

Step 3: Write the Solution in Set Notation

The inequality \(-15 < x\) tells us that the solution includes all \(x\) values greater than \(-15\). In solution set notation, we write this as: \[\{ x \,|\, x > -15 \}\]

Key Concepts

Understanding Algebra in Inequalities

Algebra allows us to work with mathematical symbols to solve various types of problems, including inequalities. Inequalities are similar to equations but instead of "equals," they use symbols like "<", ">", "≤", or "≥" to show the relationship between expressions. In the given example, \(-8 < x + 7\), we aim to find the range of values for the variable that satisfies the condition. When dealing with inequalities, it's essential to remember that they were derived from the same principles as equations:

• You can add, subtract, multiply, or divide both sides by the same number (except when multiplying or dividing by negative numbers, which flips the inequality sign).
• The key is to simplify the expression to make it easier to understand which values satisfy the inequality.

Grasping these fundamentals of algebra is crucial in resolving inequalities and is a stepping stone to mastering more complex algebraic concepts.

Deciphering Solution Set Notation

Solution set notation is a method of expressing the collection of solutions that satisfy a particular inequality. In the example, after isolating the variable, we are left with the inequality \(-15 < x\), which indicates that any number greater than \(-15\) can be a solution.To express this in solution set notation, we use curly brackets and a vertical bar, or pipe, to denote the set:

• The general form is \(\{ x \,|\, \text{condition}\}\), which translates to "the set of all \(x\) such that the condition holds true".
• In our example, the solution set \(\{ x \,|\, x > -15 \}\) tells us precisely the values \(x\) can take.

This notation is efficient for conveying complex solutions succinctly and is commonly used in mathematics to provide a clear, concise summary of variable ranges.

Techniques for Isolating Variables

Isolating variables is a fundamental process for solving inequalities or equations. This involves manipulating the expression to "get the variable by itself" on one side of the inequality or equation. In our example with \(-8 < x + 7\), the goal is to find the values for \(x\) by isolating \(x\):1. Identify what is affecting the variable. In the exercise, \(x\) is "burdened" by a \(+7\) on its right.2. Use inverse operations to simplify the expression:

• Subtract 7 from both sides to counteract the \(+7\).
• This process removes the addition of 7, resulting in \(-8 - 7 < x + 7 - 7\).
• Simplify to find \(-15 < x\), showing all values that \(x\) can take.

3. Always check if any operations change the direction of the inequality (like multiplying or dividing by a negative number).
Mastering this method is critical for solving not only straightforward inequalities but also more challenging algebraic problems.