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Magazine Chapter 8: Problem 34
Solve each double inequality. Graph the solution set and write it using interval notation. $$ -5.3 \leq x-2.3 \leq-1.3 $$
Short Answer
Expert verified
The solution set is \([-3, 1]\).
Step by step solution
01
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 \(-1.3\).
02
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:\[-3 \leq x\]
03
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:\[x \leq 1\]
04
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.
05
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.
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
- 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 \((-\infty, 3)\).
Graphing Inequalities
Graphing inequalities on a number line is a visual way to show solutions of inequalities. Let鈥檚 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.
- 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.
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 \(x \leq 1\).
- 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.
