Problem 71 - College Algebra with Modeling and Visualization

Question

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ -4 x-6>0 $$

Step-by-Step Solution

Answer

Solution: \((-\infty, -1.5)\)

Step 1: Isolate the variable term

To solve the inequality \(-4x - 6 > 0\), we first need to isolate the variable term \(-4x\). We start by adding 6 to both sides of the inequality:\[-4x - 6 + 6 > 0 + 6\]This simplifies to:\[-4x > 6\].

Step 2: Solve for the variable

Next, divide both sides of the inequality by \(-4\) to solve for \(x\). Remember that dividing by a negative number reverses the inequality sign.\[x < \frac{6}{-4}\]Simplifying the fraction on the right gives:\[x < -1.5\].

Step 3: Write the solution set

Since the inequality is \(x < -1.5\), the solution set includes all numbers less than \(-1.5\). We express this in interval notation:\[(-\infty, -1.5)\].

Key Concepts

Interval Notation

Interval notation is a way of writing sets of numbers, particularly useful when dealing with inequalities. It provides a concise method of representing all the numbers between two endpoints. In our inequality solution example, we use interval notation to express all possible values of a variable that meet a given condition.

Let's say we have the solution from our example inequality that says, "all numbers less than -1.5." The interval notation for this situation is:

• (- ∞, -1.5)

Here's how to read this interval: - The round bracket "(" indicates that -1.5 is not included in the solution but numbers approaching it can be. - The negative infinity symbol "-∞" suggests the set includes all numbers going further left without bound.

• Always remember: A round bracket "()" shows that an endpoint is not included, while a square bracket "[]" would include it.

This notation is clear and compact, making it very handy especially for graphical presentations or when dealing with larger sets of numbers.

Set-Builder Notation

Set-builder notation is another method to express sets, often used when describing the solution to an inequality. It is more descriptive and works well when the specific property of the set needs emphasis.

• In this format, you define the property that members of the set must satisfy.

In our example solution, if we achieve the inequality result "\(x < -1.5\)", we can express this using set-builder notation as:

• {\(x \mid x < -1.5\)}

Here's a breakdown of the notation:

• The curly braces "{}" indicate a set.
• The bar "|" (or "\mid") means "such that."
• The general format is: {Variable \(\mid\) Condition}.

This method allows for a clear and precise definition of the set, by focusing directly on the condition that the elements satisfy, which may be particularly useful for algebraic expressions or when explaining logical conditions.

Isolating Variables

Isolating a variable is a crucial step in solving inequalities or equations. This involves rearranging the equation so that the variable you're solving for is on one side by itself.

• The process often involves addition, subtraction, multiplication, or division to 'move' terms from one side of the equation to the other.

For the inequality \(-4x - 6 > 0\), the main goal is to isolate \(x\):

• First, add 6 to both sides to remove the constant term: \(-4x - 6 + 6 > 6\).
• The inequality simplifies to: \(-4x > 6\).
• Next, divide each side by -4. Remember, dividing or multiplying by a negative number flips the inequality sign: \(x < -\frac{6}{4}\).

Once isolated, and simplified to \(x < -1.5\), the inequality becomes much clearer.

Isolating variables involves basic algebraic manipulation and greatly aids in systematically solving problems, leading to the solution set representation.