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Magazine Chapter 2: Problem 38
For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ x-4<-2 \text { or } x-4>2 $$
Short Answer
Expert verified
The solution is \((-\infty, 2) \cup (6, \infty)\).
Step by step solution
01
Break Down the Compound Inequality
The compound inequality is given as two separate inequalities connected by 'or': \(x - 4 < -2\) and \(x - 4 > 2\). We'll solve each inequality individually for \(x\).
02
Solve the First Inequality
Start with the first inequality: \(x - 4 < -2\). Add 4 to both sides to isolate \(x\):\[x - 4 + 4 < -2 + 4\x < 2\]
03
Solve the Second Inequality
Now solve the second inequality: \(x - 4 > 2\). Again, add 4 to both sides:\[x - 4 + 4 > 2 + 4\x > 6\]
04
Analyze the 'Or' Condition
Since the compound inequality uses 'or', the solution set includes all values that satisfy either of the inequalities: \(x < 2\) or \(x > 6\).
05
Graph the Solution Set
On a number line, draw an open circle at \(x = 2\) and shade everything to the left. Also, draw an open circle at \(x = 6\) and shade everything to the right. This shows all values less than 2 or greater than 6.
06
Express the Solution in Interval Notation
The solution in interval notation is a union of two intervals: \((-\infty, 2) \cup (6, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Inequalities
To solve inequalities, you'll work to find which values of a variable will make the inequality true. In this exercise, we dealt with compound inequalities. These are two separate inequalities combined into one statement using the word "or" or "and".
- For the compound inequality with "or," as in this exercise, it means that a solution can satisfy either part of the inequality. So, we need to check both inequalities separately.
- Start by isolating the variable in each inequality. For example, with the inequality \(x - 4 < -2\), add 4 to both sides to get \(x < 2\).
- For \(x - 4 > 2\), do the same addition to both sides to reach \(x > 6\).
Interval Notation
Once you've solved the inequalities, it's important to express solutions in interval notation. Interval notation provides a concise way to describe the set of solutions on a number line.
- For example, the solution \(x < 2\) is written as the interval \((-\infty, 2)\). The parenthesis here indicates that 2 is not included in the set.
- Similarly, \(x > 6\) is expressed as \((6, \infty)\), where the parenthesis means 6 isn't a part of the solution.
- The compound solution \(x < 2\) or \(x > 6\) thus combines these intervals. We use a union symbol \(\cup\) to represent this: \((-\infty, 2) \cup (6, \infty)\).
Graphing Inequalities
Graphing inequalities is an excellent way to visually understand solution sets. A graph on a number line helps clarify which parts of the line are included in the set.
- For \(x < 2\), place an open circle on 2 to show it's not included, then shade everything to the left showing all smaller numbers are part of the solution.
- For \(x > 6\), again use an open circle on 6 and shade everything to the right, indicating all numbers greater than 6 are included.
- Combining these, the graph should highlight two separate shaded areas: one to the left of 2 and one to the right of 6.
Number Line
A number line is a simple but powerful tool for visualizing inequalities and solutions. It helps make abstract concepts tangible by showing exactly which values are included or excluded.
- When graphing on a number line, open circles are used to denote numbers that aren't part of the solution set (\(x < 2\) or \(x > 6\) in this case).
- Shading to the left or right of a circle indicates which direction represents the solution.
- The number line spans from negative to positive infinity, providing a full representation of possible values without requiring exhaustive manual listing of each one.
