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Chapter 2: Problem 61

Solve using the multiplication principle. Graph and write both set-builder notation and interval notation for each answer. $$ 1.8 \geq-1.2 n $$

Short Answer

Expert verified
The solution is \( n \geq -1.5 \ in set-builder notation, \( \{ n \ | \ n \geq -1.5 \} \ and in interval notation, \( [-1.5, \infty) \)

Step by step solution

01

Isolate the Variable

To isolate the variable, divide both sides of the inequality by -1.2. When dividing by a negative number, flip the inequality sign.
02

Perform the Calculation

Perform the division: \( \frac{1.8}{-1.2} \ Answer: -1.5 \)
03

Write New Inequality

The new inequality after flipping becomes: \( n \ greater than or equal -1.5 \)
04

Write in Set-Builder Notation

In set-builder notation, the solution is: \( \{ n \ | \ n \geq -1.5 \} \ \)
05

Write in Interval Notation

In interval notation, the solution is written as: \( [-1.5, \infty) \)
06

Graphing the Solution

On the number line, graph a solid circle at -1.5 and shade to the right to indicate all numbers greater than or equal to -1.5.

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Key Concepts

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

multiplication principle
When solving inequalities involving multiplication, you might need to use the multiplication principle. The idea is simple: you can multiply both sides of an inequality by the same number without changing the inequality's truth. However, be cautious when multiplying by a negative number. This action reverses the inequality sign. For example, in the inequality \( 1.8 \geq -1.2n \), we divide (similar to multiplying by the reciprocal) both sides by -1.2, reversing the sign to get \( n \leq -1.5 \).
set-builder notation
Set-builder notation is a concise way to describe a set of numbers satisfying a certain condition. For the inequality solution \( n \geq -1.5 \), the set-builder notation is written as:
\( \{ n \ | \ n \geq -1.5 \} \)
This notation reads as 'the set of all \( n \) such that \( n \) is greater than or equal to -1.5'. It's a mathematical shorthand to express the solution clearly.
interval notation
Interval notation provides a way to describe a range of values. It uses brackets and parentheses to show whether endpoints are included. For the inequality \( n \geq -1.5 \), the interval notation is:
\( [-1.5, \infty) \)
The square bracket [ means -1.5 is included in the solution (greater than or equal). The parenthesis ) next to infinity means infinity is not a specific endpoint but a direction. This gives a clear visual representation of the range of possible values for \( n \).
graphing inequalities
Graphing inequalities on the number line helps visualize the solution set. To graph \( n \geq -1.5 \):
  • Draw a number line.
  • Place a solid circle at -1.5 to indicate that -1.5 is included in the solution (because of the 'equal to' part).
  • Shade to the right of -1.5 to show all numbers greater than -1.5.
This shading clearly indicates the portion of the number line that is part of the solution.

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Most popular questions from this chapter

To prepare for Section 2.5, review translating to algebraic expressions and equations. Translate to an algebraic expression or equation. It has been determined that at the age of \(10,\) a girl has reached \(84.4 \%\) of her final adult height. Dana is \(4 \mathrm{ft} 8 \mathrm{in.}\) at the age of \(10 .\) What will her final adult height be?

Use an inequality and the five-step process to solve each problem. One side of a triangle is \(2 \mathrm{cm}\) shorter than the base. The other side is \(3 \mathrm{cm}\) longer than the base. What lengths of the base will allow the perimeter to be greater than \(19 \mathrm{cm} ?\)

Use an inequality and the five-step process to solve each problem. The average price of a movie ticket can be estimated by the equation \(P=0.169 Y-333.04,\) where \(Y\) is the year and \(P\) is the average price, in dollars. For what years will the average price of a movie ticket be at least \(\$ 7 ?\) (Include the year in which the \(\$ 7\) ticket first occurs.

To prepare for Section 2.5, review translating to algebraic expressions and equations. Translate to an algebraic expression or equation. The community of Bardville has 1332 left-handed females. If \(48 \%\) of the community is female and \(15 \%\) of all females are left-handed, how many people are in the community?

Use an inequality and the five-step process to solve each problem. Following the guidelines of the Food and Drug Administration, Dale tries to eat at least 5 servings of fruits or vegetables each day. For the first \(\operatorname{six}\) days of one week, she had \(4,6,7,4,6,\) and 4 servings. How many servings of fruits or vegetables should Dale eat on Saturday, in order to average at least 5 servings per day for the week?
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