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Chapter 8: Problem 37

Solve each compound inequality. Graph the solution set and write it using interval notation. $$ x \leq-2 \text { or } x>6 $$

Short Answer

Expert verified
\((-\infty, -2] \cup (6, \infty)\). Graph includes shaded regions to the left of \(-2\) and to the right of 6.

Step by step solution

01

Understand the Problem

We have a compound inequality given by \( x \leq -2 \text{ or } x > 6 \). This means the solution includes values of \( x \) that satisfy either one or both of these inequalities.
02

Solve Each Inequality Separately

First, solve \( x \leq -2 \). Any number less than or equal to \(-2\) satisfies this inequality. Next, solve \( x > 6 \). Any number greater than 6 satisfies this inequality.
03

Combine the Inequalities

Since the inequalities are joined by 'or', the solution includes all values that meet either \( x \leq -2 \) or \( x > 6 \).
04

Graph the Solution Set

On a number line, shade the region from negative infinity to -2 and include -2 (use a closed dot for \(-2\)), and then shade the region greater than 6 with an open dot (not including 6).
05

Write in Interval Notation

The interval for the solution set combines both parts: \( (-\infty, -2] \cup (6, \infty) \). '(-鈭�, -2]' indicates all numbers less than or equal to \(-2\) and '(6, 鈭�)' includes all numbers greater than 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Solving
Inequality solving involves finding all values of a variable that make an inequality true. In the given exercise, we work with a *compound inequality* that includes the word 'or'. It looks like this: \( x \leq -2 \text{ or } x > 6 \). This means we're looking for numbers that satisfy one inequality, the other, or both.
To solve the inequalities separately:
  • For \( x \leq -2 \): Solve by determining all values that are less than or equal to -2. Remember, 'less than or equal to' is often shown on a graph with a closed dot.
  • For \( x > 6 \): Solve by finding values greater than 6. Here, 'greater than' is represented with an open dot on a graph. Open dots indicate the number itself is not a part of the solution.
By solving each inequality, we determine the *solution set* for each. We then combine sets since 'or' allows for union. This means any value satisfying either inequality belongs to the solution.
Interval Notation
Interval notation is a way of expressing the set of solutions for an inequality. It鈥檚 compact and includes specific symbols to show which numbers are included in a set. Let鈥檚 explore this further using our solved inequalities:
  • For \( x \leq -2 \), in interval notation, this is \((-\infty, -2]\). The parenthesis \((-\infty\) indicates that infinity is not a number, thus never included, and the bracket \(-2]\) signifies that -2 is included.
  • For \( x > 6 \), it becomes \((6, \infty)\). Here, the parenthesis on both sides denotes that 6 is not included, and infinity naturally isn't a part of any set.
The union of these two, written as \((-\infty, -2] \cup (6, \infty)\), combines the intervals. This notation reveals two separate parts of the solution set, acknowledging that a number solves the inequality if it belongs to either interval part. Interval notation is concise, precise, and visually clear once understood.
Graphing Inequalities
Graphing inequalities on a number line gives a clear visual representation of the solution set. Let鈥檚 recap our example:
  • To graph \( x \leq -2 \): On a number line, shade the area to the left of -2. Use a closed dot at -2 to show it is included in the solution.
  • To graph \( x > 6 \): Shade the area to the right of 6 with an open dot at 6, indicating that while numbers greater than 6 are part of the solution, 6 itself is not included.
This approach helps us quickly visualize the range of solutions for both inequalities in the compound statement. With closed and open dots, we can easily distinguish included values from those that aren鈥檛. By understanding graphing techniques, students grasp how inequalities translate to real-world problems鈥攕howcasing solutions as continuous or discrete ranges, essential in mathematics and its applications.

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