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Chapter 2: Problem 31

Solve inequality. Write the solution set in interval notation, and graph it. \(-8 t>24\)

Short Answer

Expert verified
The solution set is off \( (-\infty, -3) \).

Step by step solution

01

- Isolate the Variable

To isolate the variable, divide both sides of the inequality off after checking for subtraction, addition, or multiplication options. Here, to move to the next step, divide both sides by -8:off we getoff \[ -8t > 24 \]dividing both sides by -8 we get\[ t < -3 \] (since dividing by a negative number reverses the inequality sign.)
02

- Write the Solution in Interval Notation

In interval notation, the solution set is represented by writing an interval that includes all the possible values of t. Here, t must be less than -3. The interval notation is:off \( (-\infty, -3) \).
03

- Graph the Solution

On a number line, shade all the values less than -3. Use an open circle at -3 because -3 is not included in the solution. The graph will look like this: -鈭� | ===(-鈭�)====> (- 鈭� , 鈥�3]

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Key Concepts

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

inequality solutions
Solving inequalities is a fundamental concept in algebra. It involves finding all the possible values of a variable that satisfy a given inequality. To solve an inequality like (-8t > 24) , you need to isolate the variable, just like you do when solving equations.

Here are the steps:
  • Identify the inequality sign and pay attention to its direction.
  • Add, subtract, multiply, or divide both sides of the inequality to isolate the variable.
  • Remember: when you divide or multiply both sides by a negative number, the direction of the inequality sign reverses. For instance, dividing both sides of
    -8t > 24 by (-8) , the inequality becomes (t < -3) .


This means any value of t that is less than -3 will satisfy the inequality.
interval notation
Once you've solved the inequality, you can express the solution set in interval notation. Interval notation is a shorthand way of writing subsets of real numbers and is particularly useful for describing ranges of values concisely.

Here's how to handle interval notation:
  • Use parentheses ((, )) to denote that an endpoint is not included.
  • Use square brackets ([, ]) to denote that an endpoint is included.
  • For the inequality solved above (t < -3) , the solution in interval notation is ((- , -3)) . This represents all numbers less than -3 and not including -3 .


Interval notation is especially helpful because it allows us to see the range of possible solutions at a glance.
graphing inequalities
Graphing inequalities is a visual way to represent solution sets. It helps you see which values satisfy the inequality.

To graph an inequality:
  • Draw a number line.
  • Identify the critical value that the variable cannot equal, and use an open circle if the endpoint is not included (as for < t < -3 ).
  • Shade the portion of the number line that represents all the solutions. For t < -3 , shade the line to the left of -3 .


This graphical representation clearly shows that any value less than -3 , but not including -3 , satisfies the inequality. It makes it easier to understand and communicate the solution.

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