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Chapter 4: Problem 67

Translate to an inequality. The temperature is at most \(-3^{\circ} \mathrm{C}\)

Short Answer

Expert verified
The inequality is \(T \leq\ -3^{\text{ \textdegree}} \text{C}.\)

Step by step solution

01

Understand the phrase 'at most'

The phrase 'at most' means that the value should be less than or equal to a certain number. In this case, it means the temperature should be less than or equal to \(-3^{\text{ \textdegree}} \text{C}\).
02

Identify the variable

Let the variable representing temperature be \(T\).
03

Write the inequality

Combine the variable and the condition to form the inequality:$$T \text{'at most'} \rightarrow T \leq\ -3^{\text{ \textdegree}} \text{C}.$$

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Key Concepts

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

Inequality Translation
Inequality translation in algebra involves converting words and phrases into mathematical expressions. This is crucial because it allows us to work with equations and inequalities more easily. In the problem, the phrase 'at most' tells us that the value cannot go higher than a certain point. It includes the point itself and anything lower. For example, if the temperature is 'at most' \(-3^{\circ} \text{C}\), it means the temperature can be \(-3^{\circ} \text{C}\) or any value lower than that.

The key phrases frequently found in inequality translation include:
  • 'at least' for \( \geq \)
  • 'at most' for \( \leq \)
  • 'more than' for \( > \)
  • 'less than' for \( < \)
Temperature Constraints
Temperature constraints are common in real-life problems and equations. These constraints often dictate certain conditions dependent on weather, science, or refrigeration contexts. In the exercise provided, the temperature constraint is given as 'at most \(-3^{\circ} \text{C}\).'

This means we need to consider all values that are \(-3^{\circ} \text{C}\) or lower. Understanding these constraints can be helpful not just in solving math problems but also in fields like physics, chemistry, and even everyday applications. For example, if a freezer must maintain a temperature 'at most \(-3^{\circ} \text{C}\)', this ensures items inside remain frozen.
Variable Identification
Variable identification is an essential first step in solving any algebraic problem. In this exercise, we need to identify the variable that represents the unknown quantity鈥攊n this case, temperature.

By identifying the variable (which we denote as \(T \)), we can easily combine it with the condition given to express the inequality. Here, \(T\) represents the temperature, and the inequality is written as \(T \leq \ -3^{\circ} \text{C}\).

When you identify the variables, it helps set up the equations or inequalities correctly and makes it easier to solve them. This element is foundational for tackling more complex math problems later.

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Most popular questions from this chapter

Use a graphing calculator to graph each inequality. a) \(3 x+6 y>2\) b) \(x-5 y \leq 10\) c) \(13 x-25 y+10 \leq 0\) d) \(2 x+5 y>0\)

Luggage Size . Unless an additional fee is paid, some major airlines will not check any luggage for which the sum of the item's length, width, and height exceeds 62 in. The U.S. Postal Service will ship a package only if the sum of the package's length and girth (distance around its midsection) does not exceed 130 in. Video Promotions is ordering several \(30-\) in. long cases that will be both mailed and checked as luggage. Using \(w\) and \(h\) for width and height (in inches), respectively, write and graph an inequality that represents all acceptable combinations of width and height.

Write an equivalent inequality using absolute value. $$ x<-8 \text { or } 2

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ |t|>0 $$

Write an equivalent inequality using absolute value. $$ x<-4 \text { or } 4
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