Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically. Write the solution set in interval notation. Support the answer graphically. $$8(4-3 x) \geq 6(6-4 x)$$

Step-by-Step Solution

Answer

The solution set is the empty set: \((\emptyset)\).

Step 1: Distribute terms on both sides

The inequality given is \(8(4-3x) \geq 6(6-4x)\). First, distribute the constants inside the parentheses. For the left side, distribute 8: \[8 \times 4 - 8 \times 3x = 32 - 24x\].For the right side, distribute 6: \[6 \times 6 - 6 \times 4x = 36 - 24x\].The inequality becomes \(32 - 24x \geq 36 - 24x\).

Step 2: Simplify the inequality

Now, simplify the inequality \(32 - 24x \geq 36 - 24x\) by cancelling out \(-24x\) from both sides: \[32 \geq 36\].

Step 3: Analyze the resulting statement

The statement \(32 \geq 36\) is false because 32 is not greater than or equal to 36. Therefore, there is no solution to the inequality.

Step 4: State the solution set in interval notation

Since the inequality is false for all values of \(x\), the solution set is the empty set, which can be denoted in interval notation as \((\emptyset)\).

Step 5: Support the answer graphically

Since the inequality has no solution, graphically this means that there is no overlap on the number line for any possible values of \(x\) that satisfy the inequality. Therefore, the graph would illustrate no shaded region, reflecting the empty solution set.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single number by each term inside a bracket. In simple terms, it helps in removing the parentheses, making expressions easier to simplify. When you encounter an expression like \[ a(b + c) \], the distributive property tells you that this is equivalent to \[ ab + ac \].
In the given exercise, \[ 8(4-3x) \] and \[ 6(6-4x) \] are both examples where the distributive property is used.

• For \[ 8(4-3x) \], multiply 8 by each term inside the parentheses: \[ 8 \times 4 - 8 \times 3x = 32 - 24x \].
• Similarly, for \[ 6(6-4x) \], the calculation is: \[ 6 \times 6 - 6 \times 4x = 36 - 24x \].

This distributive process is essential for simplifying and solving inequalities, as demonstrated in the problem.

Interval Notation

Interval notation is a way to describe the set of solutions for inequalities. It uses parentheses and brackets to denote intervals on the number line. Knowing how to express solutions in this way is essential in communicating your results neatly.
Interval notation is particularly useful when the solution set includes a range of numbers. For example:

• If an inequality indicates that the solutions are all numbers greater than 3, it would be written in interval notation as \((3, \infty)\).
• Alternatively, if the solution set includes all numbers less than or equal to 5, it’s written \((-\infty, 5]\).

In the exercise, the solution set is empty because the inequality has no possible solutions. Therefore, we express this lack of solution by using \((\emptyset)\), indicating that no numbers satisfy the inequality.

Graphical Representation

Graphical representation is a visual method to showcase the solution or solution sets of equations and inequalities. When dealing with inequalities, it's often useful to express these on a number line.
Here's how graphical representation applies:

• A solution set that includes real numbers can be represented as a shaded region on the number line.
• This can help in visualizing the range or interval of possible solutions easily.

In the given problem, there is no solution to the inequality \[ 32 \geq 36 \]. Since this statement is always false, the graphical representation on the number line doesn't involve any shaded regions. It visually emphasizes the emptiness of the solution set, illustrating that no values of \( x \) make the inequality true.