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Chapter 9: Problem 92

Solve the inequality and graph the solution. 8>2 x>-4

Short Answer

Expert verified
The solution to the inequality is 'x> -2'. The graphical representation shows an open circle at -2 with an arrow pointing towards positive infinity.

Step by step solution

01

Simplification

Given the inequality, 8>2 x>-4, we can simplify it further by dividing all parts by 2. This simplification leaves an inequality that's easier to work with, x>-2, which will be the basis for solving for the unknown.
02

Solve for the unknown

There's no specific step required for solving for the unknown because of the inequality's nature. According to the inequality x>-2, 'x' can be any number greater than -2.
03

Graph the solution

To graph this solution, we need a number line. We then mark -2 on this line. Since 'x' is greater than -2 (not equal), we use an open circle at -2 to represent this. Then, we draw an arrow to the right of -2 to show all the possible values of 'x'.

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Key Concepts

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

Number Line
A number line is like a picture that shows numbers in a line. It is helpful for understanding and solving inequalities.

Imagine a straight line where numbers are placed in order from left to right. Each position on this line corresponds to a real number:
  • The farther you go to the right, the larger the numbers become.
  • The farther you go to the left, the smaller they get.


Number lines usually have tick marks for integers like -3, -2, -1, 0, 1, 2, and so on. You can also include fractions and decimals if needed.

They are especially useful for comparing numbers and visualizing solutions to inequalities by showing a range of possible values.
Solutions of Inequalities
Inequalities are mathematical sentences that show the relationship between two expressions. Instead of "equals," they use symbols like ">" (greater than) and "<" (less than).

When we solve inequalities, we find all the values of the variable that make the inequality true. For example, with the inequality \(x > -2\), we want to identify all the numbers that are greater than \(-2\).

Here are some important points:
  • The solution set of an inequality is often a range of numbers, not just a single value.
  • We read \(x > -2\) as "x is greater than negative 2," meaning x can be -1, 0, 1, 2, etc.
  • The solution is depicted on the number line to visualize its range effectively.
Graphing Inequalities
Graphing inequalities on a number line is a visual way to display all possible solutions.

To graph \(x > -2\) on a number line:
  • Mark \(-2\) on the number line.
  • Since \(x\) is greater than \(-2\), place a circle over \(-2\). The circle should not be filled since \(x\) is not equal to \(-2\).
  • Draw an arrow starting from the open circle and pointing to the right, indicating all numbers greater than \(-2\).


This process creates a simple and clear picture of the infinite range of solutions for the inequality, helping to understand what values are acceptable and in which direction the solutions stretch.
Open Circle Representation
When you see an open circle on a number line, it means that number is not included in the solution. In our example, the open circle at \(-2\) shows that \(x\) is greater than \(-2\) but not equal to \(-2\).

Open circles are important for:
  • Differentiating between strict inequalities (like \(>\) and \(<\)) and inclusive ones (like \(\geq\) or \(\leq\), which use closed or filled circles).
  • Clearly indicating the starting or boundary point of a solution range without including the boundary itself.


Understanding this representation helps to correctly interpret and graph inequalities, ensuring clarity in the distinction between boundaries that are and aren't part of the solution set.

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Most popular questions from this chapter

Use linear combinations to solve the system. (Review 7.3 ) $$\begin{aligned}&12 x-4 y=-32\\\&x+3 y=4\end{aligned}$$

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{7 \pm 3 \sqrt{2}}{-1}$$

Sketch the graph of the function. Label the vertex. y=-2 x^{2}-3 x+2

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{6 \pm 4 \sqrt{2}}{-1}$$

INTERPRETING THE DISCRIMINANT Consider the equation \(\frac{1}{2} x^{2}+\frac{2}{3} x-3=0\) How many solutions does the equation have?
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