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Chapter 17: Problem 37

Graph the given inequalities on the number line. $$x<-1 \text { or } 1 \leq x<4$$

Short Answer

Expert verified
Graph \(x < -1\) with an open circle at \(-1\) and shade left; graph \(1 \leq x < 4\) with a closed circle at \(1\) and open at \(4\), and shade in between.

Step by step solution

01

Understanding the Compound Inequality

We need to graph two separate inequalities on a number line: \(x < -1\) and \(1 \leq x < 4\). These are joined by the word "or," meaning we graph both inequalities.
02

Graph \(x < -1\) on the number line

Draw an open circle at \(-1\) on the number line. Shade all numbers less than \(-1\) to the left of this point. An open circle is used because \(-1\) is not included in the solution.
03

Graph \(1 \leq x < 4\) on the number line

Draw a closed circle at \(1\) and an open circle at \(4\) on the number line. Shade the region between \(1\) and \(4\) including \(1\) but not including \(4\). A closed circle at \(1\) indicates that \(1\) is included while an open circle at \(4\) indicates that \(4\) is not included.
04

Combine the Graphs

On the number line, ensure that both sets of shading are present. This shows the solution set for \(x < -1\) or \(1 \leq x < 4\).

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Key Concepts

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

Compound Inequality
Compound inequalities deal with situations where two or more inequalities are linked by the words "and" or "or." In our exercise, the compound inequality is written as \(x < -1\) or \(1 \leq x < 4\). By using "or," we signify that a value of \(x\) can satisfy either one of these inequalities. This means we make two separate graphs and combine the solutions on a number line.
  • "Or" indicates that a solution can satisfy either inequality or both
  • Your graph will showcase multiple solution regions
This compound nature means the solution doesn't need to be a single segment on the number line. Instead, it reflects the separate ranges where the inequality conditions are valid.
Number Line
A number line is a visual tool used to represent solutions to inequalities. It helps us see which values of \(x\) satisfy the inequality. In this scenario, you'll make markings for significant points like \(-1\), \(1\), and \(4\) on the line. Then, you'll indicate which sections of the number line are part of the solution by shading them.
  • Points for important values (like boundaries) are marked distinctly
  • Shading on the number line indicates the solution sets
This allows us to visually appreciate and understand where solutions lie in relation to other numbers or inequalities on the line.
Open Circle
In graphing inequalities, an open circle is used on the number line at a boundary point that is not included in the solution set. For instance, when graphing \(x < -1\), you place an open circle at \(-1\) because values less than but not equal to \(-1\) are part of the solution.
  • Utilized to denote exclusion of boundary values in a solution set
  • Indicates the inequality does not include the exact boundary value
Open circles are critical in clearly expressing that the boundary itself does not form a valid part of the solution.
Closed Circle
A closed circle on a number line signifies that the point it represents is included in the solution of an inequality. In our specific exercise, when you graph \(1 \leq x < 4\), a closed circle is placed at \(1\) to show that \(1\) is part of the set where the inequality holds true.
  • Indicates the inclusion of the exact point in the solution set
  • Cues observers visually to acknowledge such boundary values
Using closed circles appropriately is essential to ensuring others understand fully which values are counted as valid solutions in your representation.

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