Problem 51 - College Algebra

Question

Solve compound inequality. \(6< x+3<8\)

Step-by-Step Solution

Answer

The solution of the inequality \(6< x+3<8\) is \(3 < x < 5\) or in interval notation, (3, 5)

Step 1: Subtract 3 from all parts

The goal is to isolate x. Start by subtracting 3 from all parts of the inequality. Therefore, this will become \(6 - 3 < x + 3 - 3 < 8 - 3\).

Step 2: Simplify

Now by simplifying each inequality we get \(3 < x < 5\).

Step 3: Interpret the result

This inequality represents all values of x that are larger than 3 and smaller than 5. In interval notation, this solution is represented as (3, 5)

Key Concepts

Interval Notation

Interval notation is a convenient way to represent a range of numbers on a number line. It uses brackets and parentheses to show open and closed ends of an interval.
When we have a compound inequality like \(3 < x < 5\), it means \(x\) is between 3 and 5 but does not include these values. In interval notation, we represent this as \((3, 5)\).

• Round brackets \(()\) mean the endpoints are not included (open intervals).
• Square brackets \([]\) indicate that the endpoints are included (closed intervals).

This format helps to quickly understand the range of solutions to an inequality, making it essential in algebra and calculus.

Isolate Variable

To isolate a variable means to get the variable by itself on one side of the equation or inequality. This is a critical skill in solving inequalities and equations.
In the compound inequality \(6 < x + 3 < 8\), our task is to isolate \(x\). We do this by performing the same operation on all parts of the inequality to maintain its balance.

• We subtracted 3 from each section: \(6 - 3 < x + 3 - 3 < 8 - 3\).
• This gives us the simpler inequality \(3 < x < 5\).

Isolating the variable helps us clearly see the range of values that \(x\) can take.

Inequality Solving Steps

Solving inequalities involves following specific mathematical steps to find the solution set that makes the inequality true.
Here's a breakdown of the steps you follow using the example \(6 < x + 3 < 8\):

• Step 1: Isolate the Variable - Subtract 3 from all sections to get the inequality \(3 < x < 5\).
• Step 2: Simplify - Simplify the expression to get the clean result \(3 < x < 5\).
• Step 3: Interpret - Understand what this inequality means for \(x\); it includes all numbers between 3 and 5.
• Step 4: Write in Interval Notation - Express the solution as \((3, 5)\) to convey the range succinctly.

Each step is crucial to ensure accuracy and clarity when handling inequalities.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In inequalities, understanding these expressions is essential to manipulate and solve them.
In our exercise, the expression \(x + 3\) combines the variable \(x\) and a constant number 3.
Understanding how to rearrange these terms is vital:

• The goal is to isolate \(x\) on one side of the inequality.
• By subtracting 3 from \(x + 3\), you eliminate the constant and isolate the variable.

Mastering algebraic expressions lays the groundwork for solving more complex algebraic problems in mathematics.