Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(6+3(1-x) \geq 0\) (b) \(6+3(1-x)<0\)

Step-by-Step Solution

Answer

(a) \((-\infty, 3]\), (b) \((3, \infty)\).

Step 1: Expand the Inequality

Start by expanding the inequality for part (a): \[6 + 3(1-x) \geq 0\]Distribute the 3 across the terms inside the parentheses:\[6 + 3 - 3x \geq 0\]This simplifies to:\[9 - 3x \geq 0\]

Step 2: Isolate the Variable

To isolate \(x\), subtract 9 from both sides of the equation:\[-3x \geq -9\]Next, divide both sides by -3, remembering to reverse the inequality sign when dividing by a negative number:\[x \leq 3\]

Step 3: Write the Solution in Interval Notation (Part a)

The solution for part (a) in interval notation is:\[(-\infty, 3]\]This indicates that \(x\) can be any number less than or equal to 3.

Step 4: Consider Part (b) Inequality

For part (b), we start with the related expression:\[6 + 3(1-x) < 0\]This becomes:\[9 - 3x < 0\]

Step 5: Solve the Inequality

Subtract 9 from both sides of the inequality for part (b):\[-3x < -9\]Now, divide both sides by -3, reversing the sign:\[x > 3\]

Step 6: Write the Solution in Interval Notation (Part b)

The solution in interval notation for part (b) is:\[(3, \infty)\]This indicates that \(x\) must be greater than 3.

Key Concepts

Interval Notation

Understanding interval notation is key to expressing the solution set of inequalities in a concise format. Interval notation uses brackets and parentheses to indicate the set of all numbers between two endpoints. There are two types of brackets:

• Parentheses \((\text{ or } )\): These denote that an endpoint is not included in the interval, often referred to as "open" endpoints. For example, \((3, 5)\) means all numbers greater than 3 and less than 5, but not including 3 or 5 themselves.
• Brackets \([\text{ or } ]\): These denote that an endpoint is included in the interval, known as "closed" endpoints. For instance, \([3, 5]\) includes 3 and 5 themselves, along with all numbers in between.

In our original exercise, for part (a) where the inequality solution was \(x \leq 3\), we write \((-\infty, 3]\). This shows that \(x\) can be any number from negative infinity up to and including 3. For part (b), where \(x > 3\), the solution is \((3, \infty)\), indicating all numbers greater than 3. By using interval notation, we present solutions in a clear and compact way.

Graphical Support

Graphical support helps visualize the solution of inequalities, making complex concepts easier to grasp. When graphing an inequality, a number line is often used to illustrate which values satisfy the inequality:

• Number Line Representation: Start by drawing a horizontal line. Mark important points related to the inequality, such as critical values where the inequality changes direction.
• Closed and Open Dots: Use a closed dot at an endpoint if the inequality includes that number (\(\leq\) or \(\geq\)). Use an open dot for points that are not included (\(<\) or \(>\)).
• Shading: Shade the line or area where the inequality holds true. This visually shows the range of values \(x\) can take.

For part (a) with \(x \leq 3\), place a closed dot at 3 and shade to the left. Conversely, for part (b) with \(x > 3\), use an open dot at 3 and shade to the right. These visuals reinforce the solution intervals and enhance understanding through a clear representation.

Inequality Solving

Solving inequalities involves similar steps to solving equations, but with a special rule for negative multiplication or division. Follow these steps for effective inequality solving:

• Expand and Simplify: Start by expanding any expressions using distributive properties, such as in \(6+3(1-x) \geq 0\).
• Rearrange: Collect all terms involving the variable on one side and constants on the other. This helps isolate the variable, as done by subtracting 9 from both sides to get \(-3x \geq -9\).
• Divide or Multiply: To solve for the variable, divide or multiply both sides of the inequality. Remember every time you multiply or divide by a negative number, reverse the inequality sign. So from \(-3x \geq -9\), dividing by \(-3\) changes the inequality to \(x \leq 3\).

Practicing these steps repetitively ensures accuracy and builds confidence in solving various types of inequalities. By combining these methods and understanding the rules, the solutions become straightforward to compute.