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Chapter 1: Problem 48

Write \(-\frac{1}{2} \leq z<2.4\) as two separate inequalities joined by "and."

Short Answer

Expert verified
-\frac{1}{2} \leq z \ and \ z < 2.4

Step by step solution

01

Identify the compound inequality

The given inequality is \(-\frac{1}{2} \leq z < 2.4\). This means that the variable \(z\) is bounded by two values: \(-\frac{1}{2}\) and \(2.4\).
02

Separate the compound inequality into two parts

Split the compound inequality \(-\frac{1}{2} \leq z < 2.4\) into two separate inequalities. The first part will be \(-\frac{1}{2} \leq z\) and the second part will be \(z < 2.4\).
03

Combine using 'and'

Join the two separate inequalities using the conjunction 'and': \(-\frac{1}{2} \leq z \) and \(z < 2.4\).

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Key Concepts

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

Understanding Inequalities
Inequalities are expressions used to compare two values or expressions. Unlike equations, which show equality, inequalities show that one side is either less than, less than or equal to, greater than, or greater than or equal to the other side.

Common notations include:
  • \textless: less than
  • \textgreater: greater than
  • \textleq: less than or equal to
  • \textgeq: greater than or equal to
In the given problem, the inequality \textless-\frac{1}{2} \textleq z \textless 2.4\ tells us that \(z\) lies within a range. This highlights a 'compound inequality' which shows multiple constraints on the same variable.
Inequality Notation
Inequality notation helps us to write and understand the range of values that a variable can take. In the compound inequality \(-\frac{1}{2} \textleq z \textless 2.4\), the variable \(z\) is limited to values between \(-\frac{1}{2} and 2.4\).

In expressions:
  • \textleq: indicates that \(z\) can be equal to \(-\frac{1}{2}\) or any value greater than \(-\frac{1}{2}\).
  • \textless: indicates that \(z\) is less than but not equal to \(2.4\).
When splitting and writing compound inequalities, remember to use 'and' to indicate that both conditions must be satisfied at the same time. For instance, the given compound inequality can be split into two:
  • \(-\frac{1}{2} <= z\)
  • \(z \textless 2.4\)
And they are joined by 'and'.
Basics of Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.

In inequalities, algebra helps you manipulate and solve the comparisons. Simple algebraic operations such as addition, subtraction, multiplication, and division are frequently used.

For example, solving the expression \(-\frac{1}{2} \textleq z \textless 2.4\) involves understanding how to isolate the variable \(z\) and converting the inequality into a more readable format using basic algebraic principles.

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Most popular questions from this chapter

Suppose that \(P\) dollars in principal is invested at an annual simple interest rate \(r\) for \(t\) years. Then the amount in the account \(A\) (in \$) is given by \(A=P+P r t\). With a \(4 \%\) simple interest rate, what is the minimum amount of principal required for an investment to grow to at least \(\$ 10,000\) in 8 yr? Round to the nearest cent.

Solve the equations. \(4|2 t+7|+2=22\)

A die is a six-sided cube with sides labeled with \(1,2,3,4,5,\) or 6 dots. The die is a "fair" die if when rolled, each outcome is equally likely. Therefore, the probability that it lands on " \(1 "\) is \(\frac{1}{6}\). If a fair die is rolled 360 times, we would expect it to land as a "1" roughly 60 times. Let \(x\) represent the number of times a "1" is rolled. The inequality \(\left|\frac{x-60}{\sqrt{50}}\right|<1.96\) gives the "reasonable" range for the number of times that a "1" comes up in 360 rolls. a. Solve the inequality and interpret the answer in the context of this problem. b. If the die is rolled 360 times, and a "1" comes up 30 times, does it appear that the die is a fair die?

Solve the inequality and write the solution set in interval notation. \(5 \leq|2 x+1| \leq 7\)

Write the set as a single interval. $$[(-\infty,-2) \cup(4, \infty)] \cap[-5,3)$$
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