Problem 81 - Intermediate Algebra

Question

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. \(12 x-33.16 \leq 5.84\) b. \(12 x-33.16>5.84\)

Step-by-Step Solution

Answer

Part a solution: \(( -\infty, 3.25 ]\); Part b solution: \(( 3.25, \infty )\).

Step 1: Solve the inequality in part a

Start by isolating the variable \(x\) on one side of the inequality. Given the inequality \(12x - 33.16 \leq 5.84\), add 33.16 to both sides to get \(12x \leq 39\). Then, divide both sides by 12 to solve for \(x\): \(x \leq \frac{39}{12}\). This simplifies to \(x \leq 3.25\).

Step 2: Graph the solution set for part a

To graph the solution set for \(x \leq 3.25\), draw a number line. Place a closed circle at 3.25 to indicate that 3.25 is included in the solution. Shade all the points to the left of 3.25 to represent that any number less than or equal to 3.25 satisfies the inequality.

Step 3: Write the solution set in interval notation for part a

The interval notation for the solution \(x \leq 3.25\) is \(( -\infty, 3.25 ]\). This denotes all numbers less than or equal to 3.25.

Step 4: Determine the solution for part b based on part a

The inequality in part b is \(12x - 33.16 > 5.84\). Notice that this is the opposite of the inequality in part a. Therefore, the solution set for part b can be found as the complement of part a: \(x > 3.25\). No need to redo the arithmetic.

Step 5: Graph the solution set for part b

To graph the solution set for \(x > 3.25\), draw a number line. Place an open circle at 3.25 to indicate that 3.25 is not included in the solution. Shade all the points to the right of 3.25 to represent that any number greater than 3.25 satisfies the inequality.

Step 6: Write the solution set in interval notation for part b

The interval notation for the solution \(x > 3.25\) is \(( 3.25, \infty )\). This denotes all numbers greater than 3.25.

Key Concepts

Understanding Interval Notation

Interval notation is a compact way of representing a set of numbers along a number line. It helps us express domains of inequalities effectively without listing out every single number. In interval notation, we use brackets extit{(both round () and square [])} to denote which numbers are included in or excluded from the solution set.

Here's a quick rundown:

• Round brackets \(( )\) signify that an endpoint is not included in the interval. Example: \((-\infty, 3.25)\) means that numbers up to but not including 3.25 are part of the solution set.
• Square brackets \([ ]\) signify that an endpoint is included. Example: \([-\infty, 3.25]\) would mean all numbers up to and including 3.25.
• Infinity symbols \((\pm \infty)\) are always used with round brackets because infinity is not a tangible numerical value to be included.

Interval notation simplifies communicating complex sets of numbers, particularly in solutions of inequalities and functions. By learning to interpret this notation, you can decode quite a range of mathematical statements at a glance.

Graphing Solutions of Inequalities

Graphing is a visual way of representing the solutions to inequalities, giving us a clearer picture of the range of possibilities. This approach caters to different learners who benefit from visual aids.

When you graph inequalities, first draw a horizontal number line:

• For solutions stating \(x \leq 3.25\), you mark 3.25 on the number line using a closed circle indicating it's part of the solution.
• Shade the entire region to the left of the 3.25, as all numbers in that direction satisfy the inequality.

If the solution is \(x > 3.25\):

• Use an open circle at 3.25 to show it is not part of the solution set.
• Shade the area towards the right of 3.25, indicating all greater numbers solve the inequality.

Remember, these visual cues not only help confirm your results but also serve as a quick reference for verifying calculations.

Complements in Inequalities

Complements in inequalities can simplify your mathematical problems by showing stark contrasts between solution sets. The concept of complements helps us immediately grasp solutions without redundant calculations.

For example:

• If you know the solution to \(12x - 33.16 \leq 5.84\) is \(x \leq 3.25\), the complementary inequality \(12x - 33.16 > 5.84\) immediately suggests \(x > 3.25\).
• This technique avoids repeating calculations; once you have one inequality's solution, its complement is simply the logical opposite.

By using complements, you perform a quick switch from what is included to what is excluded from a set, streamlining both problem-solving and your understanding of inequality solutions.