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Chapter 2: Problem 14

Solve each compound inequality. Graph the solution set and write it in interval notation. See Examples 2 and 3 . $$ x \leq 0 \text { and } x \geq-2 $$

Short Answer

Expert verified
The solution is \([-2, 0]\) in interval notation.

Step by step solution

01

Understanding the Compound Inequality

We begin by analyzing the compound inequality given: \( x \leq 0 \) and \( x \geq -2 \). This represents two conditions that \( x \) must satisfy simultaneously, meaning \( x \) must be less than or equal to 0 AND greater than or equal to -2.
02

Identifying the Solution Range

Next, determine which values of \( x \) satisfy both parts of the inequality. These are the values that are simultaneously \( x \leq 0 \) and \( x \geq -2 \). By combining these two inequalities, we get \(-2 \leq x \leq 0\). This means our solution range is all numbers between -2 and 0, inclusive of both -2 and 0.
03

Interval Notation

We now express this solution in interval notation. The numbers between -2 and 0, including -2 and 0 themselves, are written in interval notation as \([-2, 0]\). The square brackets always indicate that the endpoints are included in the solution set.
04

Graphing the Inequality

We then graph the solution set on a number line to visually represent the solution. Place closed dots at -2 and 0 to show these values are included, and shade the line between these points to represent all values in the interval \([-2, 0]\).

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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 concise and standardized way of representing ranges of values in mathematics. It is often used to express the solution sets of inequalities, including compound inequalities. For instance, when we solve a compound inequality like \(x \leq 0\) and \(x \geq -2\), we determine the solution range of values that satisfy both conditions simultaneously.

  • The solution range here is the set of numbers that are greater than or equal to -2 and less than or equal to 0.
  • In interval notation, this is expressed as \([-2, 0]\).
  • Square brackets \[\] indicate that the endpoints, -2 and 0, are included in the solution.
Interval notation is handy because it clearly indicates which values from the range are included or excluded, using square brackets for inclusion and parentheses for exclusion. Remembering this small detail will ensure you use interval notation confidently and correctly.
Inequality Solutions
Inequality solutions involve finding all the possible values of a variable that satisfy the given inequality condition(s). With compound inequalities, like the one given here, you must find values that satisfy more than one condition at the same time.

  • For the compound inequality \(x \leq 0\) and \(x \geq -2\), the solution requires combining both conditions into one inclusive range.
  • Firstly, determine the overlap of the ranges represented by each inequality condition.
  • Thus, we identify that all values of \(x\) between -2 and 0 satisfy both conditions, giving us the solution \(-2 \leq x \leq 0\).
When working with multiple inequalities, it is essential to analyze each condition independently first, then combine them to construct the final solution set. This process clarifies the complete picture of the possible solutions and ensures accuracy in your answer.
Graphing Inequalities
Graphing inequalities on a number line provides a visual understanding of the solution set for inequalities. This tool is especially useful for visualizing compound inequalities.

When you've determined the solution range, \([-2, 0]\) in this case, graph it to easily communicate the solution:
  • Place closed dots over the numbers -2 and 0; these closed dots indicate that both endpoints are included in the solution set.
  • Shade the line segment that connects these points to represent all the values in the range \([-2, 0]\).
By graphing, we create a simple yet powerful illustration of the inequality solution set. Graphs compliment the interval notation by providing an immediate and visual reference, making it easier to grasp which values satisfy the inequality.

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Most popular questions from this chapter

Consider the equation \(3 x-4 y=12 .\) For each value of \(x\) or \(y\) given, find the corresponding value of the other variable that makes the statement true. If \(x=4,\) find \(y\)

Solve each equation or inequality for \(x\). $$ |5-6 x|=29 $$

Determine which numbers in the set \(\\{-3,-2,-1,0,1,2,3\\}\) are solutions of each inequality. See Sections 1.4 and \(2.1 .\) $$ x+5 \leq 6 $$

Solve each inequality. Graph the solution set and write it in interval notation. $$ \left|\frac{5 x+6}{2}\right| \leq 0 $$

Write \(x>1\) or \(x<-1\) as an equivalent inequality containing an absolute value.
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