Problem 34 - Elementary and Intermediate Algebra

Question

Solve each double inequality. Graph the solution set and write it using interval notation. $$ -5.3 \leq x-2.3 \leq-1.3 $$

Step-by-Step Solution

Answer

The solution set is \([-3, 1]\).

Step 1: Understand the Double Inequality

The given double inequality is \(-5.3 \leq x - 2.3 \leq -1.3\). This means that we are trying to find all values of \(x\) such that \(x - 2.3\) is simultaneously greater than or equal to \(-5.3\) and less than or equal to \(-1.3\).

Step 2: Solve the First Inequality

First, solve \(-5.3 \leq x - 2.3\). Add 2.3 to both sides to isolate \(x\):\[-5.3 + 2.3 \leq x\]Simplifying gives:\[-3 \leq x\]

Step 3: Solve the Second Inequality

Next, solve \(x - 2.3 \leq -1.3\). Again, add 2.3 to both sides to isolate \(x\):\[x \leq -1.3 + 2.3\]Simplifying gives:\[x \leq 1\]

Step 4: Combine the Solutions

Combine the results of the two inequalities:\[-3 \leq x \leq 1\]This is the range of \(x\) that satisfies both parts of the double inequality.

Step 5: Graph the Solution Set

To graph the solution set on a number line, draw a solid line between -3 and 1 and place closed circles (or dots) at -3 and 1, indicating that these points are included in the solution set.

Step 6: Express the Solution in Interval Notation

In interval notation, the solution \(-3 \leq x \leq 1\) is expressed as:\[[-3, 1]\]This means \(x\) includes all real numbers from -3 to 1, inclusive.

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. In our exercise, the solution to the double inequality is written as the interval \([-3, 1]\). When using interval notation:

• The square brackets \([ ]\) indicate that the endpoints are included in the interval, known as "closed interval".
• The interval \([-3, 1]\) tells us that all numbers from -3 to 1 are part of the solution, including -3 and 1 themselves.

A few tips for mastering interval notation:

• If an endpoint is not included in the solution, we use parentheses \(( )\), called an "open interval". For example, \((-3, 1]\) would mean -3 is not included but 1 is.
• Make sure to write smaller numbers on the left and larger numbers on the right.
• In cases with infinity, use \(-\infty\) or \(\infty\) with open parentheses, like \((-\infty, 3)\).

Understanding these small details helps in converting inequalities and interpreting data accurately amid a problem-solving context.

Graphing Inequalities

Graphing inequalities on a number line is a visual way to show solutions of inequalities. Let’s explain the process based on our example:

• Firstly, draw a horizontal line which represents the number line.
• Mark key points, like -3 and 1, from the solution \([-3, 1]\) on this line.
• To show inclusion of these points, place solid dots or circles on -3 and 1.
• Connect these dots with a solid line to represent all numbers between -3 and 1.

This shows every number within the interval is a solution to the inequality. For different inequalities:

• If an endpoint should not be included, use an open circle. A solid line is still drawn between numbers that are solutions.
• This visual method makes it easier to understand the range of possible values quickly.

Graphing is particularly helpful for visual learners to see solution sets in another format apart from algebraic symbols.

Solving Linear Inequalities

Solving linear inequalities is a systematic process. Consider what we did in this exercise:

• The double inequality is split into two separate inequalities, \(-5.3 \leq x - 2.3\) and \(x - 2.3 \leq -1.3\).
• To solve \(-5.3 \leq x - 2.3\), add 2.3 to both sides to get \(-3 \leq x\).
• Similarly, to solve \(x - 2.3 \leq -1.3\), add 2.3 to both sides leading to \(x \leq 1\).

Combining solutions \([-3 \leq x \leq 1]\) covers the entire solution set that satisfies both original inequalities. Key steps to remember:

• Perform the same mathematical operation on both sides of an inequality, maintaining balance.
• Be cautious of flipping inequality signs when multiplying or dividing by negative numbers (not applicable here).
• Check the solution on a graph to ensure comprehension and accuracy.

By understanding these principles, you can solve not just double inequalities, but individual linear inequalities more effectively.