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Chapter 1: Problem 14

Write the given inequality using interval notation and then graph the interval. $$ x>-7 $$

Short Answer

Expert verified
Interval notation: \((-7, \infty)\); Mark an open circle at -7 and shade to the right on the number line.

Step by step solution

01

Understanding the Inequality

The inequality given is \( x > -7 \). This means that \( x \) can take any value that is greater than \(-7\), but \( x \) cannot be equal to \(-7\).
02

Convert to Interval Notation

In interval notation, the set of all numbers greater than \(-7\) is written as \((-7, \infty)\). The round parenthesis \(()\) indicates that \(-7\) is not included in the interval, and \(\infty\) always has a round parenthesis because infinity is not a number that can be reached.
03

Graph the Interval on a Number Line

To graph \((-7, \infty)\) on a number line, first put an open circle at \(-7\) to signify that \(-7\) is not included in the set. Then draw a line extending to the right (towards positive infinity) to indicate all numbers greater than \(-7\).

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Key Concepts

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

Inequalities
Inequalities are mathematical expressions used to compare two values, showcasing the relationship between them. An inequality states whether one value is less than, greater than, less than or equal to, or greater than or equal to another. For example, the inequality \(x > -7\) tells us that \(x\) should be greater than \(-7\), but not equal to it.
  • ">" means "greater than"
  • "<" means "less than"
  • ">=" means "greater than or equal to"
  • "<=" means "less than or equal to"
Inequalities are essential for determining ranges and constraints in various mathematical problems, whether in algebra, calculus, or even in real-world scenarios like budgeting or planning.
Number Line
A number line is a visual representation of numbers placed in order along a straight line. They are powerful tools for understanding and solving mathematical problems, especially when dealing with inequalities.
On a number line:
  • Numbers increase from left to right.
  • The position of a number on the line gives a visual idea of its value relative to others.
When dealing with inequalities such as \(x > -7\), the number line helps us visualize this inequality by showing an open circle at \(-7\), indicating that this value is not included, and shading or extending a line to the right beyond \(-7\) towards positive infinity.Using the number line in exercises like these helps students clearly identify which values are part of the solution and promotes a deeper understanding of inequalities and their solutions.
Graphing
Graphing in mathematics involves drawing representations of mathematical concepts to provide a visual understanding. When graphing intervals for inequalities, like \((-7, \infty)\), it's crucial to use established symbols that convey important information.
For the inequality \(x > -7\):
  • Draw an open circle at \(-7\) on the number line since \(-7\) is not part of the solution.
  • Your line should extend to the right from this open circle, representing all numbers greater than \(-7\).
The open circle signifies exclusion of the boundary point, while the arrow or line to the right represents the endless nature of numbers greater than \(-7\). Graphing inequalities in this way allows learners to grasp the range of possible solutions quickly, verifying their understanding and helping them communicate mathematical ideas effectively.

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Most popular questions from this chapter

Use rationalization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{4 y^{2}}{\sqrt{y^{2}+y+1}-\sqrt{y+1}}\) (b) \(\lim _{y \rightarrow 0} \frac{4 y^{2}}{\sqrt{y^{2}+y+1}-\sqrt{y+1}}\)

The given equation is a partial answer to a calculus problem. Solve the equation for the symbol \(y^{\prime}\). $$ y^{\prime}=2(x-y)\left(1-y^{\prime}\right) $$

Sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \((x-1)^{2}+(y+5)^{2} \leq 25\)

Sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \(1 \leq x^{2}+y^{2} \leq 4\)

The given algebraic expression is an unsimplified answer to a calculus problem. Simplify the expression. $$ \left(3 x^{2}+4 x-1\right)(4)(2 x-3)^{3}(2)+(2 x-3)^{4}(6 x+4) $$
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