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Chapter 2: Problem 12

Solve each inequality. Graph the solution set and write it in interval notation. $$ 3 x>-9 $$

Short Answer

Expert verified
The solution is \(x > -3\), or \((-3, \infty)\) in interval notation.

Step by step solution

01

Isolate the Variable

To solve the inequality \(3x > -9\), we first need to isolate \(x\). We can do this by dividing both sides of the inequality by 3, which is the coefficient of \(x\). This gives us: \[\frac{3x}{3} > \frac{-9}{3}\]Simplifying both sides, we find:\[x > -3\]
02

Graph the Solution Set

To graph the solution \(x > -3\), we draw a number line. Locate the point \(-3\) on the number line and draw an open circle around it, which represents that \(-3\) is not included in the solution set. Then, draw an arrow extending to the right, indicating all numbers greater than \(-3\) are included.
03

Write in Interval Notation

The solution \(x > -3\) in interval notation is expressed as:\((-3, \infty)\)In this notation, \(-3\) is the lower bound (not included), and \(\infty\) represents that there is no upper bound. Parentheses are used to indicate that \(-3\) is not included in the solution.

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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 concise way to express the set of all solutions for an inequality. It uses brackets and parentheses to show whether endpoints are included or excluded in an interval. For example, in the inequality solution \(x > -3\), we write the interval as \((-3, \infty)\). The parenthesis \((-3\)) means that \(-3\) is not part of the solution and is called an open interval. The infinity symbol \(\infty\) represents that there is no upper limit on \(x\), extending indefinitely towards positive infinity. Always remember:
  • Use parentheses \(()\) to represent values that are not included (open).
  • Use brackets \([]\) for values that are included (closed).
Thus, the interval notation provides a clear and efficient way to communicate the range of values \(x\) can take.
Graphing Solutions
Graphing solutions of inequalities involves visualizing the range of possible values on a number line. This is a helpful method to understand which numbers actually solve the inequality. For the inequality \(x > -3\), follow these steps:
  • First, draw a horizontal number line.
  • Identify the point that corresponds to \(-3\) on the line.
  • Since \(-3\) is not included in the solution, represent this by drawing an open circle around \(-3\).
  • Finally, draw a line or an arrow extending to the right from the open circle. This shows all numbers greater than \(-3\) are part of the solution set.
Graphing provides a visual representation, making it easier to comprehend the inequality's solution set.
Algebraic Manipulation
Algebraic manipulation involves using algebraic operations to solve for the desired variable. It is a fundamental process in solving inequalities. Let's break it down using the inequality \(3x > -9\):
  • First, identify the variable you need to solve for, which in this case is \(x\).
  • Next, to isolate \(x\), we need to eliminate the coefficient 3 beside \(x\). This is done by dividing both sides of the inequality by 3, which simplifies the inequality while maintaining its truth.
  • Perform the division: \(\frac{3x}{3} > \frac{-9}{3}\) results in \(x > -3\).
By simplifying both sides, we make it clear that any number greater than \(-3\) satisfies the original inequality. Always perform operations across the entire inequality to preserve the relationship between the variable and the numbers involved.

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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.

To "break even "in a manufacturing business, revenue \(R\) (income) must equal the cost \(C\) of production, or \(R=C\). The cost \(C\) to produce \(x\) number of skateboards is given by \(C=100+20 x .\) The skateboards are sold wholesale for \(\$ 24\) each, so revenue \(R\) is given by \(R=24 x .\) Find how many skateboards the manufacturer needs to produce and sell to break even. (Hint: Set the expression for \(R\) equal to the expression for \(C,\) then solve for \(x .\) )

Solve each inequality. Graph the solution set and write it in interval notation. $$ x^{2}-4 x+8

Match each equation in the first column with its solution in the second column. Items in the second column may be used more than once. Explain the difference between simplifying an expression and solving an equation.

Evaluate the following. $$ (3)^{3} $$
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