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Chapter 4: Problem 41

Graph and write interval notation for each compound inequality. $$ 5>a \text { or } a>7 $$

Short Answer

Expert verified
The solution is \((-\infty, 5) \cup (7, \infty)\).

Step by step solution

01

- Identify the Inequalities

There are two inequalities given: \(5 > a\) or \(a > 7\).
02

- Rewrite the Inequalities in Standard Form

Rewrite each inequality in standard form for clarity: \(a < 5\) or \(a > 7\).
03

- Graph the Inequalities on a Number Line

Draw a number line and mark points at 5 and 7. Shade to the left of 5 to represent \(a < 5\) and shade to the right of 7 to represent \(a > 7\). Use open circles at 5 and 7 because these values are not included in the inequalities.
04

- Write the Solution in Interval Notation

Combine the regions from the graph to write the solution in interval notation: \((-\infty, 5) \cup (7, \infty)\).

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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 represent the relationship between two values where one is less than, greater than, less than or equal to, or greater than or equal to the other. In this exercise, we are working with strict inequalities: \(5 > a\) and \(a > 7\).

For \(5 > a\), we can rewrite it in its standard form as \(a < 5\). This means that \(a\) can be any value less than 5.

For \(a > 7\), \(a\) is any value greater than 7.

Notice the use of inequality symbols:
  • \( < \) means less than
  • \( > \) means greater than
  • \( eq \) means not equal to
  • \(eqeq \) means combination of both
Whenever you encounter inequalities, make sure to translate them into a form you are comfortable with and can understand easily.
number line graphing
Number line graphing is an excellent visual method to understand and solve inequalities. It helps you see which parts of the number line are included in the solution.

For the inequalities \(a < 5\) and \(a > 7\):
  • Draw a horizontal line with evenly spaced points representing numbers.
  • Mark points at 5 and 7.
  • Shade the region to the left of 5 to represent \(a < 5\).
  • Shade the region to the right of 7 to represent \(a > 7\).
Use open circles at 5 and 7. This indicates that the values 5 and 7 themselves are not included in the solution set. Open circles are used because the inequalities are strict (do not include equals).

Number line graphs not only help visualize the scope of the variable but also help combine multiple inequalities intuitively.
interval notation
Interval notation is a concise way of writing subsets of the real number line. It uses intervals to describe the set of all possible values that satisfy an inequality.

For \(a < 5\) or \(a > 7\): We write these regions as two open intervals and combine them with the union symbol \(\bigcup\).
  • The interval \((-\infty, 5)\) represents all values less than 5.
  • The interval \( (7, +\infty\)) represents all values greater than 7.
  • Together, we write them as \((-\infty, 5) \bigcup (7, +\infty)\).
Interval notation provides a clear and concise way to represent multiple intervals and is widely used in mathematical expressions. When combined with graphs, it offers a powerful way to visualize and solve inequalities.

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Most popular questions from this chapter

solve $$ |3 x-5|=x $$

Let \(f(x)=5+|3 x+2| .\) Find all \(x\) for which \(f(x)<19\)

College Faculty. The number of part-time instructional faculty in U.S. postsecondary institutions is growing at a greater rate than the number of full-time faculty. The number of parttime faculty, in thousands, is approximated by $$ p(t)=27 t+325 $$ and the number of full-time faculty, in thousands, is approximated by $$ f(t)=16 t+500 $$ For both functions, \(t\) represents the number of years after \(1995 .\) Using an inequality, determine those years for which there were more part-time faculty than full-time faculty.

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