Problem 33
Question
\(33-38\) . Express the interval in terms of inequalities, and then graph the interval. $$ (-3,0) $$
Step-by-Step Solution
Verified Answer
The interval in inequality form is
-3 < x < 0. It is graphed with open circles at -3 and 0 on a number line.
1Step 1: Understand the Interval Notation
The given interval
(-3, 0)
means that we are considering all numbers between -3 and 0, not including -3 and 0 themselves.
2Step 2: Express the Interval in Inequality Form
To express the interval (-3, 0) in terms of inequalities, we note that -3 is the lower bound and 0 is the upper bound. Since the interval is open, it is represented as:\[-3 < x < 0.\]
3Step 3: Prepare to Graph the Interval
On a number line, we will mark points at
-3
and
0.
These are not included in the solution, so we will use open circles to indicate that the endpoints are not part of the interval.
4Step 4: Graph the Interval
Draw a number line. Mark open circles at
-3
and
0.
Draw a line connecting these open circles. The shaded portion between these two points represents all numbers
x
such that
-3 < x < 0.
Key Concepts
InequalitiesOpen IntervalNumber Line GraphAlgebra
Inequalities
Inequalities are a way of expressing the relative size or order of two values. When we talk about inequalities in mathematics, we often use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to compare values. For instance, if we say \(-3 < x < 0\), it means that the variable \(x\) can take any value more than \(-3\) but less than \(0\).
Using inequalities helps us in defining a set of numbers that satisfy certain conditions. They are particularly useful in algebra, where we solve equations to find unknowns. Here, inequalities help determine the range or domain in which the variable holds the true value.
Understanding how to express and solve inequalities allows students to handle a variety of mathematical problems with different limitations and boundaries.
Using inequalities helps us in defining a set of numbers that satisfy certain conditions. They are particularly useful in algebra, where we solve equations to find unknowns. Here, inequalities help determine the range or domain in which the variable holds the true value.
Understanding how to express and solve inequalities allows students to handle a variety of mathematical problems with different limitations and boundaries.
Open Interval
An open interval is a part of the real number line which includes all the numbers lying between two given numbers, but does not include the endpoints themselves. In mathematical notation, an open interval is expressed with parentheses. For example, the notation \((-3, 0)\) is an open interval that includes all numbers between \(-3\) and \(0\), excluding the numbers \(-3\) and \(0\) themselves.
The absence of the endpoint values distinguishes open intervals from closed intervals, which are represented using square brackets (e.g., \([-3, 0]\)). This concept is crucial when considering continuity, limits, and defining functions within specific bounds in calculus or real analysis.
The absence of the endpoint values distinguishes open intervals from closed intervals, which are represented using square brackets (e.g., \([-3, 0]\)). This concept is crucial when considering continuity, limits, and defining functions within specific bounds in calculus or real analysis.
Number Line Graph
A number line graph provides a visual representation of numbers and intervals, making it easier to comprehend the concepts of greater than and less than along with the relationships between different numbers.
When graphing an open interval like \((-3, 0)\), you would:
When graphing an open interval like \((-3, 0)\), you would:
- Draw a straight horizontal line that represents the number line.
- Mark the points \(-3\) and \(0\) on this line.
- Use open circles at \(-3\) and \(0\) to indicate that these points are not included in the interval.
- Shade the portion of the line between \(-3\) and \(0\) to show all numbers included in this interval.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In algebra, we often work with equations and inequalities involving variables such as \(x\). Algebraic expressions connect with interval notation when expressing solutions to inequalities.
For example, solving an inequality may result in an interval notation that represents the set of possible solutions, like \(-3 < x < 0\). This use of algebra enables us to solve real-world problems by forming equations or inequalities and looking for the values that satisfy them. Understanding how to manipulate these expressions is foundational for further study in mathematics, including calculus and other higher-level fields.
For example, solving an inequality may result in an interval notation that represents the set of possible solutions, like \(-3 < x < 0\). This use of algebra enables us to solve real-world problems by forming equations or inequalities and looking for the values that satisfy them. Understanding how to manipulate these expressions is foundational for further study in mathematics, including calculus and other higher-level fields.
Other exercises in this chapter
Problem 33
\(29-46\) Simplify each expression. $$ w^{-2} w^{-4} w^{6} $$
View solution Problem 33
29-34 . Factor the expression by grouping terms. $$ x^{3}+x^{2}+x+1 $$
View solution Problem 33
\(29-38=\) Simplify the expression. Assume that the letters denote any real numbers. $$ \sqrt[3]{x^{3} y} $$
View solution Problem 33
Find the sum, difference, or product. \(r\left(r^{2}-9\right)+3 r^{2}(2 r-1)\)
View solution