Problem 73 Solve each compound inequality, ... [FREE SOLUTION]

Chapter 8: Problem 73

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ -6

Short Answer

Expert verified

The solution is $1 < x \leq 3$ or interval notation: $(1, 3]$.

Step by step solution

Set up Compound Inequality

The compound inequality is given as $-6 < f(x) \leq 0$, where $f(x) = 3x - 9$. We will set up two separate inequalities to solve for $x$: $-6 < 3x - 9$ and $3x - 9 \leq 0$.

Solve the First Inequality

To solve $-6 < 3x - 9$, add 9 to both sides to isolate the term with $x$: $-6 + 9 < 3x$, which simplifies to $3 < 3x$. Next, divide both sides by 3: $1 < x$ or $x > 1$.

Solve the Second Inequality

For the inequality $3x - 9 \leq 0$, add 9 to both sides: $3x \leq 9$. Then, divide by 3 to solve for $x$: $x \leq 3$.

Combine the Inequalities

Combine the solutions from the separate inequalities: $x > 1$ and $x \leq 3$. This can be expressed as $1 < x \leq 3$.

Graph the Solution Set

Graph the inequality on a number line. Open circle at 1 (because $x$ is not equal to 1) and closed circle at 3 (because $x$ can be equal to 3). Shade the region between these points to represent $1 < x \leq 3$.

Write in Interval Notation

Express the solution set in interval notation. Since $1 < x \leq 3$, the solution set is $(1, 3]$.

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Key Concepts

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

Inequality Notation

Inequality notation is a mathematical way of expressing that one value is larger or smaller than another value. It uses symbols like "<" for "less than," ">" for "greater than," "鈮�" for "less than or equal to," and "鈮�" for "greater than or equal to."

In the exercise, the compound inequality is initially expressed as $-6 < f(x) \leq 0$. This includes two parts:

$-6 < f(x)$: This means that $f(x)$ is greater than $-6$.
$f(x) \leq 0$: This indicates that $f(x)$ is less than or equal to $0$.

Through solving, the compound inequality becomes $1 < x \leq 3$, effectively narrowing down the range of $x$ that satisfies both parts of the inequality.

This notation is particularly helpful for visualizing what values are included within a set of solutions that satisfy given conditions.

Number Line Graphing

Graphing solutions on a number line is an effective way to visually represent the range of solutions to an inequality.

For our solved inequality, $1 < x \leq 3$, we need to show all $x$-values that fall strictly greater than 1 and up to and including 3.

To graph this:

Place an open circle at 1 on the number line. This signifies $x$ cannot be exactly 1. An open circle indicates that the endpoint is not part of the solution.
Place a closed circle at 3, meaning $x$ can be exactly 3. A closed circle suggests that this point is included in the solution.
Shade the number line between 1 and 3 to show all the possible values of $x$ between these points.

This graphical representation aids in quickly understanding which values are included in the solution set and is especially useful when working with compound inequalities.

Interval Notation

Interval notation provides a concise way to denote subsets of real numbers that represent the solution to inequalities.

For the inequality $1 < x \leq 3$, the interval notation is written as $(1, 3]$.

Here's how it works:

The round parenthesis "(" at the beginning indicates that the endpoint $x = 1$ is not included in the solution, corresponding to an open circle on the number line.
The square bracket "]" at the end means that the endpoint $x = 3$ is included, corresponding to a closed circle on the number line.
This notation effectively communicates the range of values $x$ can take, allowing for quick and clear interpretation of solution sets.

Interval notation simplifies the writing of solutions for compound inequalities, making it easier to read and understand without needing a drawn-out explanation of the set boundaries.

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91影视

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ -6

Short Answer

Step by step solution

Set up Compound Inequality

Solve the First Inequality

Solve the Second Inequality

Combine the Inequalities

Graph the Solution Set

Write in Interval Notation

Key Concepts

Inequality Notation

Number Line Graphing

Interval Notation

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