Problem 50 - College Algebra

Question

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$7

Step-by-Step Solution

Answer

The solution to the inequality \(7 < x+5 < 11\) is \( 2 < x < 6 \). This is graphed on a number line by marking the range between 2 and 6. The solution expressed in interval notation is (2 , 6).

Step 1: Isolation of the Variable

First, the inequality \(7 < x+5 < 11\) needs to be written in a way where the variable, x, is isolated. To isolate x, subtract 5 from all parts of the inequality to get \(2 < x < 6\).

Step 2: Graphing the Solution Set

Next is to graph the solution set on a number line. On a number line, denote the range from 2 to 6. Both the points will be open points as they are not included in the solution set. The sections on the number line going from left of '2' and right of '6' will be left blank, as those sections are not within the solution set. The range between '2' and '6' will be colored or marked to show all the numbers in between are part of the solution set.

Step 3: Express the Solution Set in Interval Notation

Finally, the solution set will be expressed in interval notation. Since '2' and '6' are not included in the solution set, parentheses will be used. Moreover, the symbol 'U' denotes a union and keeps the solution sets separate. Here there is only one solution set, so there is no need for 'U'. As a result, the interval notation would be (2 , 6).

Key Concepts

Interval Notation

Interval notation is a way of expressing the set of numbers, especially when dealing with inequalities. This notation helps to describe the range of values that a variable can assume. It is simple and straightforward, making it a favorite in mathematics. The notation uses parentheses and square brackets to convey the range.

• Parentheses, like \((a, b)\), indicate that the endpoints are not included in the set.
• Brackets, like \([a, b]\), indicate that the endpoints are included.
• If an endpoint is a positive or negative infinity, you always use a parenthesis, because infinity is a concept, not a number.

For example, the inequality \(2 < x < 6\) means that \(x\) lies between 2 and 6, but it does not include 2 and 6 themselves. Therefore, the interval notation for this inequality is \((2, 6)\). This notation clearly communicates that you can pick any number between 2 and 6, but not 2 or 6.

Number Line Graphing

Number line graphing is a visual way to represent solutions to inequalities. It's a helpful tool that gives a clear picture of the range of numbers involved.To graph an inequality like \(2 < x < 6\), follow these steps:

• Draw a horizontal line, this represents the number line.
• Mark the key numbers from the inequality, in this case, '2' and '6'.
• Since '2' and '6' are not included in the solution (as indicated by the inequality signs), use open circles (not filled) at these points.
• Shade or draw a line between the open circles to show that every number between '2' and '6' is a part of the solution.

This graph provides a quick overview of where the variable values lay, automatically excluding invalid parts and helping to confirm the solution visually.

Solving Inequalities

To solve linear inequalities, the main goal is to isolate the variable just like in an equation. However, inequalities have special rules, especially concerning multiplication or division by negative numbers, which may change the inequality direction.Consider the inequality given: \(7 < x + 5 < 11\). Here’s how to solve it step by step:

• First, perform operations to isolate the variable. In this case, subtract 5 from all parts of the inequality to simplify it to \(2 < x < 6\).
• This isolates \(x\), revealing the range of solutions \(x\) can assume.
• Always pay attention to the inequality signs. If you multiply or divide the inequality by a negative number, you need to flip the inequality sign to correctly represent the solution.

Inequalities, much like equations, require balanced operations to maintain the truth of each part. Solving linear inequalities focuses on finding a range of values rather than a single number, showcasing the idea of continuous solutions.