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Chapter 2: Problem 16

Solve each inequality. Graph the solution set and write it in interval notation. $$ -5 x<20 $$

Short Answer

Expert verified
The solution is \(x > -4\), or in interval notation, \((-4, \infty)\).

Step by step solution

01

Isolate the variable

To solve the inequality \(-5x < 20\), we want to isolate \(x\). To do this, divide both sides of the inequality by \(-5\). Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. So,\[ x > \frac{20}{-5} \] Simplifying this gives,\[x > -4\]
02

Write in Interval Notation

The solution \(x > -4\) means all numbers greater than negative four. In interval notation, this is expressed as \((-4, \infty)\).
03

Graph the Solution Set

To graph the solution set \(x > -4\), plot an open circle at \(x = -4\) on the number line to indicate that \(-4\) is not included in the solution. Then, draw a line extending to the right, toward infinity, to indicate all numbers greater than \(-4\).

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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 way to represent the set of all numbers that satisfy an inequality. It uses brackets and parentheses to show which endpoints are included or excluded. In our problem, the solution to the inequality \(x > -4\) involves all numbers greater than equal to ot equal to \(-4\).
To represent this in interval notation:
  • Use a parenthesis "\((\)" if the number is not included (like with strict inequalities > or <).
  • Use a bracket "\([\)" if the number is included (like with inclusive inequalities \(\geq\) or \(\leq\)).
Since \(-4\) is not part of the solution set, we write the solution as \((-4, \infty)\).
Note that infinity always gets a parenthesis since it is not a specific number and cannot be included in any set.
Graphing Inequalities
Graphing inequalities visually represents the range of numbers that solve the inequality on a number line. For our solution \(x > -4\), graphing it makes understanding the solution much simpler. Here's how you can do it:
  • First, identify the critical point, which is \(-4\) in this case. You plot an open circle at \(-4\) to show that it is not included in the solution set.
  • Next, draw a line extending to the right of \(-4\), moving towards infinity. This represents all numbers that are greater than \(-4\), as the solution involves all such numbers.
Remember, using an open circle or a closed dot on the number line conveys whether the endpoint is included. Open circles denote excluded points, while closed dots would be for included points.
Algebraic Manipulation
Algebraic manipulation involves the process of rearranging and simplifying mathematical expressions or equations. In the context of inequalities, the goal is often to isolate the variable on one side. Here鈥檚 how it was done in the example:
  • The inequality started as \(-5x < 20\). To isolate \(x\), you need to divide both sides by \(-5\). Remember, dividing or multiplying both sides by a negative number flips the inequality sign. In our case, it flips from < to >.
  • Once the inequality is manipulated correctly, \(x > -4\) is achieved. Now, the inequality is simplified, providing us with an easy-to-understand solution.
Be cautious with the rules of inequalities, especially when working with negative numbers. Correctly applying these rules ensures accurate solutions every time.

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Most popular questions from this chapter

To "break even "in a manufacturing business, revenue \(R\) (income) must equal the cost \(C\) of production, or \(R=C\). Exercises 41 through 44 involve finding the break-even point for manufacturing. Discuss what happens if a company makes and sells fewer products than the break-even point. Discuss what happens if more products than the break-even point are made and sold.

Twice a number, increased by one, is between negative five and seven. Find all such numbers.

In 1971, T. Theodore Fujita created the Fujita Scale (or F Scale), which uses ratings from 0 to 5 to classify tornadoes, based on the damage that wind intensity causes to structures as the tornado passes through. This scale was updated by meteorologists and wind engineers working for the National Oceanic and Atmospheric Administration (NOAA) and implemented in February 2007 . This new scale is called the Enhanced \(\mathrm{F}\) Scale. An EF-0 tornado has wind speeds between 65 and \(85 \mathrm{mph}\), inclusive. The winds in an EF-1 tornado range from 86 to \(110 \mathrm{mph}\). In an EF- 2 tornado, winds are from 111 to \(135 \mathrm{mph}\). An EF- 3 tornado has wind speeds ranging from 136 to \(165 \mathrm{mph} .\) Wind speeds in an EF- 4 tornado are clocked at 166 to \(200 \mathrm{mph}\). The most violent tornadoes are ranked at EF-5, with wind speeds of at least 201 mph. (Source: Storm Prediction Center, NOAA) Letting \(y\) represent a tornado's wind speed, write a series of six inequalities that describe the wind speed ranges corresponding to each Enhanced Fujita Scale rank.

To "break even "in a manufacturing business, revenue \(R\) (income) must equal the cost \(C\) of production, or \(R=C\). Find the break-even quantity for a company that makes \(x\) number of computer monitors at a cost \(C\) given by \(C=870+70 x\) and receives revenue \(R\) given by \(R=105 x\)

Perform the indicated operations. See Sections 1.5 and 1.6 $$ \frac{3}{4}-\frac{3}{16} $$
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