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Chapter 2: Problem 92

Write inequality in interval notation, and graph the interval. \(6 \geq x \geq-4\)

Short Answer

Expert verified
The interval notation is \([-4, 6]\).

Step by step solution

01

- Understand the inequality

The given inequality is expressed as a compound inequality: \(6 \geq x \geq-4\) signifies that \(x\) must be less than or equal to \(6\) and greater than or equal to \(-4\).
02

- Write in interval notation

In interval notation, \([a, b]\) signifies that \(x\) is between \(a\) and \(b\), inclusive. Since \(x\) is between \(-4\) and \(6\), including both endpoints, the interval notation is \([-4, 6]\).
03

- Graph the interval

Draw and label a number line. Place a solid dot at \(-4\) and a solid dot at \(6\). Then, shade the region on the number line between \(-4\) and \(6\). This indicates all values between \(-4\) and \(6\), 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 describe sets of numbers lying between two endpoints. It's particularly useful for expressing the solutions to inequalities. In interval notation, brackets and parentheses are used to show whether endpoints are included or excluded:
  • Brackets [ ]: Indicate that the endpoint values are included (closed interval). For example, \( [ -4, 6 ] \) means \( - 4 \leq x \leq 6 \).
  • Parentheses ( ): Indicate that the endpoints are not included (open interval). For example, \( ( -4, 6) \) means \( - 4 < x < 6 \).

For the inequality \(6 \geq x \geq -4 \), we see that both endpoints, -4 and 6, are included in the solution. Therefore, we use brackets: \[ -4 , 6 \] . This tells us that every number from -4 to 6, including -4 and 6, is a solution to the inequality.
number line
A number line is a visual representation of numbers arranged in sequential order. It's useful for graphing solutions to inequalities. Here's how you can plot the given interval \[ -4, 6 \] on a number line:
  • Draw a horizontal line and mark numbers at equal intervals.
  • Identify the points -4 and 6 on the number line.
  • Place a solid dot at -4 and another solid dot at 6. Solid dots indicate that these points are included in the interval.
  • Shade the region between -4 and 6. This shaded part represents all the numbers between these two points, inclusive of -4 and 6.

Using a number line helps visually confirm the solution to the inequality and ensures a clear understanding.
compound inequality
A compound inequality combines two or more simple inequalities joined by 'and' (\(\cap\)) or 'or' (\(\cup\)). In the given problem, we have an 'and' compound inequality: \( 6 \geq x \geq-4 \). This signifies that \( x \) must satisfy both the inequality \( x \leq 6 \) and the inequality \( x \geq-4 \) at the same time.
Here鈥檚 a step-by-step look at how we get to our final solution:
  • Break down into two simple inequalities: \( x \leq 6 \) and \( x \geq-4 \).
  • Analyze each inequality: any number satisfying \( x \leq 6 \) must be less than or equal to 6, and any number satisfying \( x \geq-4 \) must be greater than or equal to -4.
  • Combine the solutions to get the numbers that satisfy both conditions simultaneously: this is every number from -4 to 6 inclusive.

Understanding compound inequalities is crucial for solving more complex algebra problems, and the use of number lines and interval notation makes this understanding clearer.

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Most popular questions from this chapter

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