Problem 70

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -4 \leq-13(3 x+12)<4 $$

Step-by-Step Solution

Verified

Answer

The solution set is $(-\frac{160}{39}, -\frac{152}{39}]$, in interval notation: $(-4.10, -3.90]$.

1Step 1: Break Down the Double Inequality

The given inequality is \[-4 \leq -13(3x+12) < 4\].This means you have two inequalities that you need to solve separately: 1. $-4 \leq -13(3x+12)$2. $-13(3x+12) < 4$.

2Step 2: Solve the First Inequality

Solve $-4 \leq -13(3x+12)$.First, distribute the $-13$ into the parenthesis:$-4 \leq -39x - 156$.Add 156 to both sides:$152 \leq -39x$.Divide both sides by $-39$, remembering to reverse the inequality:$- \frac{152}{39} \geq x$.Approximate $- \frac{152}{39} \approx -3.90$.

3Step 3: Solve the Second Inequality

Now solve $-13(3x+12) < 4$.Distribute the $-13$ into the parenthesis:$-39x - 156 < 4$.Add 156 to both sides:$-39x < 160$.Divide both sides by $-39$, remembering to reverse the inequality:$x > -\frac{160}{39}$.Approximate $-\frac{160}{39} \approx -4.10$.

4Step 4: Combine the Inequality Solutions

Combine the results from Step 2 and Step 3: $-\frac{160}{39} < x \leq -\frac{152}{39}$.This represents the solution set in inequality notation.

5Step 5: Express Solution in Interval Notation

The solution set can be expressed as an interval by using the values found:$(-\frac{160}{39}, -\frac{152}{39}]$.And approximately:$(-4.10, -3.90]$.

6Step 6: Graph the Solution

To graph, plot on a number line:- Draw an open circle at approximately $-4.10$ to indicate that it is not included in the set.- Draw a closed circle at approximately $-3.90$ to indicate that it is included in the set.- Shade the region between the two points to represent all the numbers that satisfy the inequality.

Key Concepts

Understanding the Solution SetInterval Notation DescribedVisualizing with the Number LineGraphing the Solution Set

Understanding the Solution Set

A **solution set** is a collection of all the values that satisfy a given inequality. In this exercise, the inequality is split into two separate parts to find the values of $x$ that make it true:

For $-4 \leq -13(3x+12)$, the solution is $x \leq -\frac{152}{39}$.
For $-13(3x+12) < 4$, the solution is $x > -\frac{160}{39}$.

The solution set is where both conditions are met. This results in $-\frac{160}{39} < x \leq -\frac{152}{39}$. It's important to note that the solution includes $-\frac{152}{39}$ but excludes $-\frac{160}{39}$. This captures all the values that $x$ can take to satisfy both inequalities.

Interval Notation Described

When we use **interval notation**, we're presenting the solution set as a range of values. This notation helps articulate precisely where numbers fall in relation to each other, especially:

Brackets $[\ ]$ indicate a number is included in the set.
Parentheses $(\ )$ indicate a number is not included.

For this inequality, the interval notation is $(-\frac{160}{39}, -\frac{152}{39}]$. Here:

$(-\frac{160}{39}$ shows $-\frac{160}{39}$ is excluded.
$[-\frac{152}{39}]$ means $-\frac{152}{39}$ is included.

This conveys that $x$ can be any number greater than $-\frac{160}{39}$ and up to $-\frac{152}{39}$.

Visualizing with the Number Line

A **number line** is a great tool to visually represent the solution set. It makes it clear and straightforward:

Identify $-4.10$ (for $-\frac{160}{39}$) and place an open circle there.
Locate $-3.90$ (for $-\frac{152}{39}$) and place a closed circle there.

The open circle at $-4.10$ shows that it is not included, while the closed circle at $-3.90$ shows that it is part of the solution set.
Shade the line between these two points to represent all the values that satisfy the inequality.