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Chapter 1: Problem 1

Write the given statement as an inequality. $$ a+2 \text { is positive } $$

Short Answer

Expert verified
The inequality is \(a + 2 > 0\).

Step by step solution

01

Understanding the Statement

The statement given is 'a + 2 is positive'. This means that the expression \(a + 2\) is greater than zero.
02

Translate the Statement into an Inequality

To express that 'a + 2' is positive, we write it as an inequality: \(a + 2 > 0\).
03

Verify the Inequality

Check if the inequality correctly represents the given statement. When \(a + 2\) is greater than 0, it aligns with the term 'is positive', meaning \(a + 2 > 0\) is the correct translation.

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Key Concepts

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

Positive numbers
A positive number is a number that is greater than zero. You can find positive numbers on the right side of zero on a number line. They can be whole numbers like 1, 2, 3, as well as fractions and decimals like 0.5 or 3.75.
Positive numbers are important for indicating gains or increases. They are used in a variety of everyday scenarios, including measuring heights, accounting for profits, and expressing quantities.
  • Examples of positive numbers: 5, 42, 0.001
  • They are opposite of negative numbers, which are less than zero.
  • To check if a number is positive, simply verify if it's more than zero.
Understanding positive numbers is fundamental in translating mathematical expressions and writing inequalities, as these numbers often indicate the "greater than" relationship in inequalities.
Mathematical expressions
Mathematical expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. For example, the expression \(a + 2\) consists of a variable \(a\), the number 2, and the addition operation.
Expressions can represent complex relationships and are used to solve equations and understand inequalities. They are the building blocks of mathematical logic and help us to model real-world situations in a mathematical format.
  • Key parts: Numbers, variables, and operations
  • Example: \(3x - 4\) where 3 is a coefficient, \(x\) is a variable, and -4 is a constant term
  • An expression becomes meaningful when you perform operations or solve it concerning an equation or inequality.
Knowing how to construct and deconstruct mathematical expressions is crucial, especially when we need to convert them into inequalities by specifying a condition like positiveness.
Translating statements into inequalities
Translating a statement into an inequality involves interpreting the meaning of the words to form a mathematical expression. For example, the phrase "a + 2 is positive" means that the sum \(a + 2\) is greater than zero. It translates into the inequality \(a + 2 > 0\).
Inequalities help express a range of possible values that variables can take. They are particularly useful in identifying bounds or limits and are used widely in both algebra and real-world applications.
  • Steps: Identify the mathematical expression in the statement.
  • Interpret clues like 'is positive', 'is at least', etc., to determine the inequality sign.
  • Rewrite the statement using the mathematical inequality.
  • Example translation: '5 less than a number is at least 3' translates to \(x - 5 \geq 3\).
Practicing these translations deepens understanding of mathematical language and empowers problem-solving in algebraic contexts.

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Most popular questions from this chapter

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\) (b) \(\lim _{x \rightarrow 5} \frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=\frac{(x+2)(x-8)}{x+1}\)

In Problems \(41-66\), find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=-3 x\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=\frac{x^{2}-7}{x^{3}}\)

Complete the square in \(x\) and \(y\) to find the center and the radius of the given circle. \(\frac{1}{2} x^{2}+\frac{1}{2} y^{2}+\frac{5}{2} x+10 y+5=0\)
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