Problem 43 For Problems $35-44$, solve ea... [FREE SOLUTION]

Chapter 2: Problem 43

For Problems $35-44$, solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ 3 x+2<-1 \text { or } 3 x+2>1 $$

Short Answer

Expert verified

The solution is $(-\infty, -1) \cup \left(-\frac{1}{3}, \infty\right)$.

Step by step solution

Identify Each Inequality

The problem presents two separate inequalities: $3x + 2 < -1$ and $3x + 2 > 1$. We need to solve each inequality individually before combining the results using 'or'.

Solve the First Inequality

Start with the inequality $3x + 2 < -1$. Subtract 2 from both sides to isolate the term with $x$: $3x < -3$. Next, divide both sides by 3 to solve for $x$: $x < -1$.

Solve the Second Inequality

Now solve the inequality $3x + 2 > 1$. Subtract 2 from both sides to isolate $3x$: $3x > -1$. Then divide both sides by 3 to solve for $x$: $x > -\frac{1}{3}$.

Combine the Solutions Using 'Or'

Since the original compound inequality uses 'or', the solution is the union of the two solutions. Combine them: $x < -1$ or $x > -\frac{1}{3}$.

Express in Interval Notation

Translate the inequality solutions into interval notation. The solutions are $(-\infty, -1) \cup \left(-\frac{1}{3}, \infty\right)$.

Graph the Solution Sets

On a number line, shade the regions corresponding to $x < -1$ and $x > -\frac{1}{3}$. Both parts have open circles at $-1$ and $-\frac{1}{3}$, indicating that these points are not included in the solution.

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Key Concepts

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

Interval Notation

Interval notation is a simplified way to describe solution sets of inequalities. It lets us express ranges of values and show whether endpoints are included. There are a few symbols to know:

Parentheses ( ): Indicate that an endpoint is not included.
Brackets [ ]: Indicate that an endpoint is included.
The symbol $\infty$ or $-\infty$: Represents unbounded intervals; these are always paired with parentheses because infinity and negative infinity aren鈥檛 actual numbers we can "reach".

For example, the interval (-\infty, -1) represents all numbers less than $-1$, but not including -1 itself. While the interval [-2, 3) includes all numbers from $-2$ to less than $+3$. This notation is painless once you get the hang of it, and it makes inequalities easy to work with and visualize!

Union of Sets

The union of sets is a fundamental concept in both algebra and set theory. When we take the union of two sets, we combine all distinct elements from each set into a new single set. In our problem, we were working with inequalities combined by the logical 'or', meaning any value satisfying at least one inequality is included in the solution.

The notation $A \cup B$ represents the union of sets $A$ and $B$.
In terms of intervals, the union helps us express multiple solution sets at once. For example, if the solution set from solving inequalities results in the intervals (-\infty, -1) and (-\frac{1}{3}, \infty), the union of these intervals is written as (-\infty, -1) \cup (-\frac{1}{3}, \infty).

This expression shows all numbers less than \-1\ or greater than \-\frac{1}{3}\, effectively merging the two ranges into a complete solution.

Solving Inequalities

Solving inequalities is a lot like solving equations. The main goal is to isolate the variable, often represented by $x$. However, with inequalities, there鈥檚 an important rule to remember: if you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign.

First, perform operations like addition or subtraction to isolate terms involving $x$.
Next, use multiplication or division to get $x$ by itself.
Always simplify your inequality after each step to keep calculations correct.

In our example, solving $3x + 2 < -1$ involved subtracting \-2\ and dividing by 3. A similar process was used for $3x + 2 > 1$. These steps led us to the final expressions $x < -1$ and $x > -\frac{1}{3}$. Accurate handling of inequalities helps ensure these mistakes are avoided.

Graphing Inequalities

Graphing inequalities can visually demonstrate the solution set on a number line. It helps confirm that your solutions from inequalities indeed match what they represent.

Draw a number line and mark critical points, in this case, \-1\ and \-\frac{1}{3}\.
Use open circles on essential points if those points aren鈥檛 included in the solution. This happens when inequalities use < or >.
Shade the regions representing the solutions鈥攍eft for $x < -1$ and the right for $x > -\frac{1}{3}$.

For our exercise, these shaded regions quickly show the segments of the number line representing valid solutions to the inequalities. This visual guide is handy, especially when dealing with multiple inequalities or complex solution sets, enhancing our overall understanding of the compound inequality.

91影视

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ 3 x+2<-1 \text { or } 3 x+2>1 $$

Short Answer

Step by step solution

Identify Each Inequality

Solve the First Inequality

Solve the Second Inequality

Combine the Solutions Using 'Or'

Express in Interval Notation

Graph the Solution Sets

Key Concepts

Interval Notation

Union of Sets

Solving Inequalities

Graphing Inequalities

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