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Chapter 4: Problem 29

Graph and write interval notation for each compound inequality. $$ 1

Short Answer

Expert verified
Interval notation: (1, 3). Graph: Open circles at 1 and 3 with shading in between.

Step by step solution

01

Understand the inequality

The given inequality is compound, meaning it has two parts: \( 1 < x \) and \( x < 3 \). This inequality states that \( x \) is between 1 and 3 but does not include 1 and 3.
02

Write the interval notation

Interval notation represents the set of all numbers between two endpoints. For \( 1 < x < 3 \), the interval notation is \( (1, 3) \) because 1 and 3 are not included in the interval.
03

Graph the inequality

To graph \( 1 < x < 3 \), draw a number line and put open circles at 1 and 3 to show that these numbers are not included. Then, shade the region between 1 and 3. This shaded part represents all numbers that satisfy the inequality.

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Key Concepts

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

Inequality Notation
Inequality notation is a way to express the range of values that a variable can take. It uses the signs <, >, 鈮�, and 鈮�. Specifically, < means 'less than' and > means 'greater than'. For example, the notation 1 < x < 3 describes all values of x that are greater than 1 and less than 3. It's like saying, 'x is somewhere between 1 and 3, but not including 1 and 3'. Understanding this helps you figure out what numbers make the inequality true.
Interval Notation
Interval notation is a shorthand way to write ranges of values. Unlike inequality notation, it uses parentheses and brackets:
  • Parentheses ( ) indicate that the endpoints are not included.
  • Brackets [ ] indicate that the endpoints are included.
For instance, the interval notation for the compound inequality 1 < x < 3 is (1, 3). This tells us that x can be any number between 1 and 3, but it cannot be exactly 1 or exactly 3. Points are excluded when parentheses are used. Remember, use parentheses for 'less than' and 'greater than' inequalities, and brackets for 'less than or equal to' and 'greater than or equal to' inequalities.
Graphing Inequalities
Graphing inequalities on a number line visually represents the range of possible values for a variable. Here's how to graph the compound inequality 1 < x < 3:
  • First, draw a horizontal line to represent the number line.
  • Next, find and mark the points 1 and 3 on this line.
  • Put an open circle at 1 and 3 to show that these points are not included in the solution.
  • Shade the region between these two points. This shaded part shows all the numbers that satisfy the inequality.
By doing this, you translate the inequality into a visual format, making it easier to understand. It shows at a glance where the variable's values lie.

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Most popular questions from this chapter

Solve. $$ \begin{aligned} 2 x-3 y &=5 \\ x+3 y &=-1 \end{aligned} $$

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ |t|>0 $$

Elevators. Many elevators have a capacity of 1 metric ton \((1000 \mathrm{kg}) .\) Suppose that \(c\) children, each weighing \(35 \mathrm{kg},\) and \(a\) adults, each \(75 \mathrm{kg}\), are on an elevator. Graph a system of inequalities that indicates when the elevator is overloaded.

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ 25-2|a+3|>19 $$

Explain why the inequality \(|x+5| \geq 2\) can be interpreted as "the number \(x\) is at least 2 units from \(-5 . "\)
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