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Chapter 5: Problem 70

Write an inequality to represent the statement. The difference of \(x\) and \(y\) is not less than 4 .

Short Answer

Expert verified
\( x - y \, \geq \, 4 \)

Step by step solution

01

Understand the Statement

The statement says the difference between two variables, x and y, is not less than 4. This means that the difference is either equal to 4 or greater than 4.
02

Express the Difference Mathematically

The difference between two variables, x and y, is written as: \[ x - y \]
03

Formulate the Inequality

Given that the difference is not less than 4, it can be written as an inequality: \[ x - y \, \geq \, 4 \]
04

Combine the Steps

Putting it all together, the inequality to represent the statement 'The difference of x and y is not less than 4' is: \[ x - y \, \geq \, 4 \]

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Key Concepts

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

difference
In mathematics, the term 'difference' generally refers to the result of subtracting one number from another. In the given exercise, we illustrate the difference between two variables, denoted as x and y. When we subtract y from x, we express this mathematically as:

  • \( x - y \)
Here, x represents what is being subtracted from, and y is the subtrahend. The operation x - y provides us the value that tells us how much x exceeds y, or falls short if x is smaller. For example, if x = 7 and y = 5, the difference \( x - y \) would be 2. This simple concept is fundamental in understanding more complex mathematical operations and plays a crucial role in forming inequalities.
inequality symbols
Inequality symbols are critical in math for showing relationships between values that are not necessarily equal. The primary inequality symbols are:

  • Greater than: \(>\)
  • Less than: \(<\)
  • Greater than or equal to: \(\geq\)
  • Less than or equal to: \(\leq\)
In the context of our exercise, we use the 'greater than or equal to' symbol \( \geq \). This symbol indicates either a greater value or equality. For instance, when we state \( x \geq y \), it means x is either greater than y or exactly equal to y. Inequality symbols help in describing a range of possible values in mathematical expressions and are extensively used in algebra, calculus, and beyond.
greater than or equal to
The 'greater than or equal to' symbol \( \geq \) combines the concepts of both greater than and equality. When we use this symbol, it includes two possibilities:
  • The first value can be greater than the second value.
  • The first value can be exactly equal to the second value.
For example, if we write \( x - y \geq 4 \), this means:
  • x - y can be greater than 4 (for example, 5, 6, etc.).
  • x - y can be exactly equal to 4.
In our specific exercise, 'The difference of x and y is not less than 4' can be translated using the 'greater than or equal to' symbol. This accurately reflects that the difference (\( x - y \)) is either 4 or more than 4. This symbol is powerful in constraints and optimization problems, ensuring solutions meet required conditions or constraints.

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Most popular questions from this chapter

Use substitution to solve the system for the set of ordered triples \((x, y, \lambda)\) that satisfy the system. $$ \begin{array}{l} 2=2 \lambda x \\ 6=2 \lambda y \\ x^{2}+y^{2}=10 \end{array} $$

A paving company delivers gravel for a road construction project. The company has a large truck and a small truck. The large truck has a greater capacity, but costs more for fuel to operate. The load capacity and cost to operate each truck per load are given in the table. $$ \begin{array}{|l|c|c|} \hline & \text { Load Capacity } & \text { Cost per Load } \\ \hline \text { Small truck } & 18 \mathrm{yd}^{3} & \$ 120 \\ \hline \text { Large truck } & 24 \mathrm{yd}^{3} & \$ 150 \\ \hline \end{array} $$ The company must deliver at least 288 yd \(^{3}\) of gravel to stay on schedule. Furthermore, the large truck takes longer to load and cannot make as many trips as the small truck. As a result, the number of trips made by the large truck is at most \(\frac{3}{4}\) times the number of trips made by the small truck. a. Determine the number of trips that should be made by the large truck and the number of trips that should be made by the small truck to minimize cost. b. What is the minimum cost to deliver gravel under these constraints?

Jonas performed an experiment for his science fair project. He learned that rinsing lettuce in vinegar kills more bacteria than rinsing with water or with a popular commercial product. As a follow-up to his project, he wants to determine the percentage of bacteria killed by rinsing with a diluted solution of vinegar. a. How much water and how much vinegar should be mixed to produce 10 cups of a mixture that is \(40 \%\) vinegar? b. How much pure vinegar and how much \(40 \%\) vinegar solution should be mixed to produce 10 cups of a mixture that is \(60 \%\) vinegar?

The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration \(A_{0}\) of the drug in the bloodstream at the time of injection. However, the physician knows that after \(3 \mathrm{hr}\), the drug concentration in the blood is \(0.69 \mu \mathrm{g} / \mathrm{dL}\) and after \(4 \mathrm{hr}\), the concentration is \(0.655 \mu \mathrm{g} / \mathrm{dL}\). The model \(A(t)=A_{0} e^{-k t}\) represents the drug concentration \(A(t)\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) in the bloodstream \(t\) hours after injection. The value of \(k\) is a constant related to the rate at which the drug is removed by the body. a. Substitute 0.69 for \(A(t)\) and 3 for \(t\) in the model and write the resulting equation. b. Substitute 0.655 for \(A(t)\) and 4 for \(t\) in the model and write the resulting equation. c. Use the system of equations from parts (a) and (b) to solve for \(k .\) Round to 3 decimal places. d. Use the system of equations from parts (a) and (b) to approximate the initial concentration \(A_{0}\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) at the time of injection. Round to 2 decimal places. e. Determine the concentration of the drug after \(12 \mathrm{hr}\). Round to 2 decimal places.

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{r} 3 x+y=6 \\ x+\frac{1}{3} y=2 \end{array} $$
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