Problem 15 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 4 x-1<\frac{3-x}{-3} $$

Step-by-Step Solution

Answer

\(x\) is in the interval \((-\infty, 0)\).

Step 1: Eliminate the Negative Denominator

Multiply both sides of the inequality by -3. Since we're multiplying by a negative number, the inequality direction flips.\(-3(4x - 1) > 3 - x\)This becomes:\(-12x + 3 > 3 - x\)

Step 2: Combine Like Terms

Add \(12x\) to both sides to get the x terms on one side.\(3 > 3 + 11x\)

Step 3: Solve for x

Subtract 3 from both sides to isolate the term with x:\(0 > 11x\)Now, divide by 11 to solve for x:\(x < 0\)

Step 4: Write the Solution in Interval Notation

The solution \(x < 0\) can be expressed in interval notation as:\((-\infty, 0)\)

Key Concepts

Set-builder notation

Set-builder notation is a method for describing a set by specifying a property that its members must satisfy. In mathematics, when we solve inequalities, we often use this notation to express the solution set. It tells us exactly which elements belong to the set. For example, if we found that our solution is all numbers less than 0, in set-builder notation, we would write this as:

• \(\{ x \mid x < 0 \}\)

This means "the set of all \(x\) such that \(x\) is less than 0." Using set-builder notation can be very helpful to clearly communicate the criteria any number must meet to be considered part of the solution to an inequality.

Interval notation

Interval notation is another way to express the solution set of an inequality. It provides a clear and concise way to list all the real numbers that are solutions to the inequality. For a solution indicating all numbers less than 0, interval notation would represent it as:

• \((-\infty, 0)\)

This notation uses parentheses and square brackets to show open and closed ends of an interval. If the boundary number is not included, like in \((-\infty, 0)\), we use parentheses. However, if the boundary is included, we use a bracket, such as \([-5, 2]\). This method is particularly useful in calculus and other advanced mathematics.

Solving inequalities

Solving inequalities involves finding the set of all possible values of the variable that satisfy the inequality. In the given exercise, you start by simplifying the inequality step-by-step:

• Eliminate fractions or negative denominators if necessary, remembering critical rules such as flipping the inequality sign when multiplying or dividing both sides by a negative number.
• Combine like terms to consolidate variables on one side.
• Isolate the variable on one side of the inequality to find its possible values.

Solving these steps correctly ensures you determine the range of possible values that are solutions, making it a vital skill in algebra and real-world applications.

Algebraic manipulation

Algebraic manipulation is a powerful tool that lets you rearrange and simplify expressions to solve equations and inequalities. It involves operations such as adding, subtracting, multiplying, and dividing both sides of an equation or inequality. Here's the process as seen in our example:

• Multiply through by a negative number, remembering to flip the inequality sign: transforming \(-3(4x - 1) > 3 - x\) into \(-12x + 3 > 3 - x\).
• Reorganize terms by adding \(12x\) to both sides, leading to \(3 > 3 + 11x\).
• Isolate \(x\) by subtracting numbers and ultimately dividing by a positive number, resulting in \(x < 0\).

Mastering these skills is essential for accurately interpreting and solving mathematical problems. With practice, algebraic manipulation becomes an intuitive process that aids in tackling various math challenges.