Problem 99 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically. Write the solution set in interval notation. Support the answer graphically. $$-4(3 x+2) \geq-2(6 x+1)$$

Step-by-Step Solution

Answer

The solution set is \(\emptyset\), meaning no solution exists for the inequality.

Step 1: Expand Both Sides

Start by expanding the expressions on both sides of the inequality. For the left-hand side: \(-4(3x+2) = -12x - 8\) and for the right-hand side: \(-2(6x+1) = -12x - 2\). The inequality becomes \(-12x - 8 \geq -12x - 2\).

Step 2: Simplify the Inequality

Since the term \(-12x\) appears on both sides of the inequality, they can be canceled out. This simplifies the inequality to \(-8 \geq -2\).

Step 3: Compare the Constants

Examine the resulting inequality \(-8 \geq -2\). Since this statement is false (because \(-8\) is not greater than or equal to \(-2\)), it indicates that there is no value of \(x\) that satisfies the inequality.

Step 4: State the Solution in Interval Notation

As the inequality results in a false statement, the solution set is empty, which in interval notation is represented as \(\emptyset\) or an empty set.

Key Concepts

Interval Notation

Interval notation is a mathematical way of expressing a set of numbers along an interval. It's particularly useful for describing solutions to inequalities. For instance, if an inequality results in all numbers greater than a certain value, interval notation allows us to express this range concisely.

Here are some key points about interval notation:

• Intersections: A solution set can be expressed with parentheses ")" or brackets"]". Parentheses are used when values are not included, and brackets are used when they are.
• Infinite numbers: Use the symbols "+∞" and "-∞" when the solution extends indefinitely in either direction.
• Empty set: If no solutions exist, as in our exercise, the solution set is denoted as \(ackslashemptyset\).

Interval notation provides a clear and structured way to communicate solutions, making it easier for both writing and reading mathematical solutions.

Graphical Representation

Graphically representing inequalities provides a visual perspective on the solution set. Typically, we use a number line or a coordinate plane to show where solutions lie. This can help to quickly identify which values of the variable make the inequality true.

In our particular problem, the inequality simplifies to \(-8 \geq -2\). Since this inequality is always false, it is reflected in a graphical representation by simply noting that no solutions exist. Hence, nothing appears on the number line past marking it as an empty set.

When dealing with more complex inequalities that have solutions, shading or marking certain sections of the line can clearly display the solution intervals. You typically use an open dot to signify that a number is not included in the range and a closed dot when a number is included.

Analytical Methods

Analytical methods for solving inequalities involve algebraically manipulating the inequality until the variable of interest is isolated or simplified. Let's explore this with the given exercise.

1. **Expand terms**: Initially, distribute any parentheses across both sides. In our case, we had expressions such as \(-4(3x+2)\) which needed expanding.
2. **Combine like terms**: Once expanded, group and simplify similar terms on either side of the inequality.
3. **Simplify**: This includes canceling out identical terms on both sides. As seen with \(-12x\) terms in the example, this step can substantially simplify our work.
4. **Evaluate constants**: The example's core simplification involved checking constants \(-8 \geq -2\). Since it's false, it implied no solution for the inequality.

These methods require careful attention to sign changes and validity of operations applied in each step. Understanding these concepts thoroughly allows for effective problem-solving in inequality scenarios.