Problem 10 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -4 x-3<5 $$

Step-by-Step Solution

Answer

The solution set is \(x > -2\) or \((-2, \infty)\).

Step 1: Isolate the Variable Term

To solve the inequality \(-4x - 3 < 5\), we begin by isolating the term that includes the variable \(x\). Add 3 to both sides of the inequality:\[-4x - 3 + 3 < 5 + 3\] which simplifies to \[-4x < 8\].

Step 2: Solve for the Variable

Now, to solve for \(x\), divide each side of the inequality by \(-4\). Remember, dividing by a negative number reverses the inequality sign. Thus, we have:\[x > \frac{8}{-4}\] which simplifies to \[x > -2\].

Step 3: Express the Solution Set

The solution to the inequality \(-4x - 3 < 5\) is \(x > -2\). In interval notation, this is expressed as \((-2, \, \infty)\).

Key Concepts

Set-Builder Notation

Set-builder notation is a mathematical notation used to describe a set by stating the properties that its members must satisfy. It allows us to write sets in a concise way, specifying the condition that elements of the set should fulfill. For the solution of the inequality \(-4x - 3 < 5\), we isolated the variable and solved for \(x\), which gave the result \(x > -2\).
In set-builder notation, the solution set to this inequality is expressed as:

• \( \{ x \mid x > -2 \} \)

This reads as "the set of all \(x\), such that \(x\) is greater than \(-2\)." It effectively tells us the rule or the condition the elements of the set are subject to. Using set-builder notation is advantageous when we need to clearly express the condition that dictates membership in the set.

Interval Notation

Interval notation is a method of writing sets of numbers, particularly subranges of the real number line. It is commonly used to represent solutions to inequalities, as it is both compact and reader-friendly. In our solution, we have that \(x > -2\).
To express this range in interval notation:

• We write it as \((-2, \infty)\).

Here's how to understand the components:

• \((-2, \infty)\): The round bracket or parenthesis around \(-2\) means \(-2\) is not included, aligning with \(x > -2\).
• \(\infty\) is used to indicate that there is no upper limit. Infinity always receives a parenthesis because it is not a number you can "reach," just a concept showing continuation.

This format is extremely useful in mathematics to easily communicate the continuous nature of the solution, making it clear at a glance where the values begin and end.

Inequality Reversal When Dividing by Negative

When solving inequalities, one critical rule needs special attention: the inequality sign must be flipped when both sides are divided by a negative number. This rule keeps the order of the numbers correct.
Let's take the example from the exercise, where we have \(-4x < 8\). When dividing each side by \(-4\) to solve for \(x\), the inequality sign flips, turning \(<\) into \(>\). Thus, we end up with \(x > -2\).
This reversal can seem counterintuitive, but it makes sense because of how number lines work with negative numbers:

• Think of a simple comparison: \(-3 < -1\). If you multiply both numbers by \(-1\), you get \(3 > 1\), and see how the relationship changes.

This concept is fundamental when working with any inequality, ensuring the mathematical relationships between numbers remain true despite the sign change.