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Chapter 1: Problem 18

Write the given interval as an inequality. $$ [-5, \infty) $$

Short Answer

Expert verified
The inequality is \(x \geq -5\).

Step by step solution

01

Analyze the Interval

Given the interval \([-5, \infty)\), we note that it includes numbers starting from -5 and extending indefinitely toward positive infinity. The square bracket \([-5\) indicates that -5 is included in the interval, while the parenthesis at \(\infty)\) indicates that infinity is not a specific endpoint.
02

Translate to Inequality Form

To express the interval \([-5, \infty)\) as an inequality, we need to state that the variable (usually \(x\)) can be any number greater than or equal to -5. This is expressed as the inequality \(x \geq -5\).

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Key Concepts

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

Interval Notation
Interval notation provides a concise way of expressing a range of numbers on the real number line. It uses brackets and parentheses to denote whether endpoints are included or not.
  • A square bracket [ or ] means the endpoint is included, often called "closed."
  • A parenthesis ( or ) means the endpoint is not included, called "open."
The given interval \[[-5, \infty)\]indicates all numbers starting at -5 and extending to infinity, including -5 but not infinity itself. In scenarios where you encounter an infinity symbol, it is always accompanied by a parenthesis because infinity is conceptual and cannot be reached or included.
Inequality Notation
Inequality notation is a mathematical way of expressing that two values are not necessarily equal. Instead, it shows the relative size or order of two values, and tells us if one value is greater or less than another. Examples of inequality symbols include:
  • \(<\) means "less than"
  • \(>\) means "greater than"
  • \(\leq\) means "less than or equal to"
  • \(\geq\) means "greater than or equal to"
The interval \([-5, \infty)\) can be written in inequality notation as \(x \geq -5\). This means any number, denoted by \(x\), is greater than or equal to -5.
Number Line
The number line is a visual representation of all real numbers in a straight line. It allows us to easily understand and visualize concepts such as intervals and inequalities. Each point on the number line corresponds to a real number.
  • Positive numbers are located to the right of zero.
  • Negative numbers are located to the left of zero.
  • The number line extends indefinitely in both directions.
When considering the interval \([-5, \infty)\), we can visualize it on the number line by shading from -5 all the way to the right, indicating that it includes all numbers greater than or equal to -5. -5 would appear as a filled-in point to show inclusion.
Real Numbers
Real numbers encompass all the numbers that you work with on an everyday basis. They include whole numbers, fractions, decimals, and irrational numbers. They form a continuous line of values with no gaps.
  • Whole numbers and integers: {0, 1, 2, -3, etc.}
  • Rational numbers: numbers that can be expressed as fractions, for example, \(\frac{1}{2}\) or \(3.75\).
  • Irrational numbers: numbers that cannot be exactly expressed as a fraction, like \(\pi\) or \(\sqrt{2}\).
When dealing with intervals like \([-5, \infty)\), we are considering all real numbers starting from -5 and continuing without bound in the positive direction.

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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) \(\lim _{x \rightarrow 5} \frac{x^{4}-5 x^{3}+4 x-20}{x^{4}-5 x^{3}+x-5}\)

In Problems \(31-34\), sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \(x^{2}+y^{2} \geq 9\)

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. \(y=|x-9|\)

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. \(y=2-\sqrt{x+5}\)

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. \(y=x^{2}-4\)
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