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Chapter 8: Problem 60

Graph the solution set, and write it using interval notation. $$ -4<\frac{2}{3} x<12 $$

Short Answer

Expert verified
The solution set is \(-6 < x < 18\) and in interval notation: \((-6, 18)\).

Step by step solution

01

- Solve the Inequality on the Left

First, isolate the variable in the left inequality by multiplying all parts by the reciprocal of the fraction. Multiply \[-4<\frac{2}{3} x\] by \(\frac{3}{2} \) to get \(-4 \times \frac{3}{2} < x \.\) Simplifying this gives \[-6 < x\].
02

- Solve the Inequality on the Right

Next, isolate the variable in the right inequality by again multiplying all parts by the reciprocal of the fraction. Multiply \(\frac{2}{3} x < 12\) by \(\frac{3}{2} \) to get \( x < 18\).
03

- Combine the Inequalities

Now combine the results of both inequalities: \[-6 < x < 18\] to form the complete solution.
04

- Graph the Solution

To graph the solution, draw a number line from -10 to 20. Place open circles at -6 and 18, since these values are not included in the solution set. Shade the region between -6 and 18.
05

- Write in Interval Notation

Express the solution set as \((-6, 18)\) in interval notation where the parentheses indicate that the endpoints -6 and 18 are not included.

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Key Concepts

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

graphing inequalities
Graphing inequalities visually represents the range of solutions on a number line. This method helps you better understand and verify the solution set to an inequality problem.

To graph \(-6 < x < 18\), you'll follow these steps:
  • Draw a number line that includes the numbers -6 and 18. Extend it a bit past these points for clarity.
  • Place an open circle at -6 and another one at 18. The open circles show that these endpoints are not part of the solution.
  • Shade the region between -6 and 18 to indicate that all the numbers in this region satisfy the inequality.

Graphing inequalities on a number line helps you visually confirm that all values between -6 (exclusive) and 18 (exclusive) work within the inequality.
interval notation
Interval notation is a shorthand way of writing the set of solutions to an inequality.

In interval notation, the solution set \(-6 < x < 18\) is written as \((-6, 18)\). The parentheses indicate that -6 and 18 are not included in the set, unlike brackets \[ \] which include endpoints.

Here are some basic rules for interval notation:
  • Use \(a, b\) for open intervals where endpoints a and b are not included.
  • Use \[a, b\] to include endpoints a and b in the interval.
  • Use \[a, b\) or \(a, b\] if only one endpoint is included.
  • For unbounded intervals, use 鈭� or -鈭� with parentheses, like \((-鈭�, a)\) or \(b, 鈭瀄).

Understanding interval notation allows you to communicate solution sets effectively and is crucial in algebra and calculus.
number line
A number line is a visual tool for representing numbers and their relationships. It is especially useful when solving and graphing inequalities.

  • Draw a horizontal line and mark evenly spaced values. Make sure your line is long enough to represent the set of numbers in your inequality.
  • Highlight specific points to show important values like boundaries or solutions.
  • Use open or closed circles to indicate whether endpoints are included in the solution set.

For the inequality \(-6 < x < 18\), your number line will:
  • Feature -6 and 18 marked clearly.
  • Have open circles at -6 and 18, showing they are not included in the solution.
  • Be shaded between -6 and 18 to show the range of numbers that satisfy the inequality.

Using a number line simplifies understanding and validating solutions, making them a valuable tool in algebra.

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Most popular questions from this chapter

Graph the solution set of each system of linear inequalities. $$ \begin{array}{r} 2 x+3 y<6 \\ x-y<5 \end{array} $$

Solve each compound inequality. Graph the solution set, and write it using interval notation. \(3 x+2 \leq-7\) or \(-2 x+1 \leq 9\)

Graph the solution set, and write it using interval notation. $$ -1

Graph each linear inequality. $$ x+y>0 $$

Solve each compound inequality. Graph the solution set, and write it using interval notation. \(2 x-6 \leq-18\) and \(2 x \geq-18\)
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