Problem 12 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 3(x+5) \leq 0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -5]\) in interval notation.

Step 1: Distribute the 3

Start by distributing the 3 across the expression inside the parentheses. This means multiplying 3 by both terms inside the parentheses: \[ 3(x + 5) = 3x + 15 \] So the inequality becomes: \[ 3x + 15 \leq 0 \]

Step 2: Isolate the variable term

Next, we need to isolate the term with the variable, 3x. To do this, subtract 15 from both sides of the inequality:\[ 3x + 15 - 15 \leq 0 - 15 \]This simplifies to:\[ 3x \leq -15 \]

Step 3: Solve for x

Now we need to solve for \( x \). Divide both sides of the inequality by 3:\[ \frac{3x}{3} \leq \frac{-15}{3} \]This simplifies to:\[ x \leq -5 \]

Step 4: Express the solution

We have determined that \( x \leq -5 \). We can express this solution in interval notation as follows:The solution set is:\[ (-\infty, -5] \] This means \( x \) includes all real numbers less than or equal to -5.

Key Concepts

Solving Inequalities

In the realm of algebra, solving inequalities plays a vital role in understanding how different quantities compare to one another. The goal is to find all possible values of a variable that satisfy the inequality condition. Here's a breakdown of the process:

• Distribute and Simplify: Start by simplifying the expression, just like in the given inequality, where 3 gets distributed over the terms inside the parentheses, simplifying it to \(3x + 15\).


• Isolate the Variable: The next step involves isolating the variable on one side of the inequality, similar to isolating \(3x\) by subtracting 15 from both sides, resulting in \(3x \leq -15\).


• Solving for the Variable: Dividing or multiplying through by a constant to solve for the variable is crucial here. For our case, dividing by 3 yields \(x \leq -5\).

Remember, when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips its direction. However, this wasn't necessary in this example.

Set-Builder Notation

Set-builder notation is one of the ways to neatly express a solution set of an inequality. It provides a clear, mathematical format to describe all the possible solutions.When we use set-builder notation, we specify a set and a condition that members of that set must satisfy. For example, in the problem where we've solved \(x \leq -5\), we can express this in set-builder notation as:\[ \{ x \mid x \leq -5 \} \]Here's how to read and understand this:

• The curly brackets \( \{ \} \) show we are defining a set.


• \( x \) represents the variable.


• The vertical bar, \( \mid \), is read as "such that". It separates the variable from its condition, effectively saying "the set of all \( x \) such that...."


• \( x \leq -5 \) is the condition that specifies what values make up the set.

This notation is very powerful for communicating solutions clearly and concisely.

Interval Notation

Interval notation is another elegant method to express solutions of inequalities. It uses intervals to show a range of values the variable can take, often more visually intuitive than set-builder notation.For the inequality \(x \leq -5\), its corresponding interval notation is:\[ (-\infty, -5] \]Here's what each part means:

• The round bracket \((\) before \(-\infty\) implies that \(-\infty\) is never included in a set because infinity isn't a specific number.


• The square bracket \([\) after \(-5\) indicates that \(-5\) is included in the solution set; thus, \(x\) can be exactly \(-5\) or anything smaller.

Interval notation is compact and very handy in various mathematical scenarios, especially when dealing with large ranges of numbers. It's important to interpret brackets correctly to avoid misunderstandings about the inclusivity or exclusivity of endpoints.