Problem 45 - Algebra and Trigonometry

Question

In all exercises, other than \(\varnothing\). use interval notation to express solution sets and graph each solution set on a mumber line. In Exercises \(27-50\), solve each linear inequality. $$ \frac{x-4}{6} \geq \frac{x-2}{9}+\frac{5}{18} $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 15\).

Step 1: Simplify the fractions

Combine the fractions on the right side of the equation -> \(\frac{x-2}{9}+\frac{5}{18} = \frac{2(x-2)+5}{18}\)

Step 2: Ensure common denominators

In order to combine the left side of the equation with the right side, we need to ensure both sides have common denominators. Multiply the left side by 3 -> \(3*\frac{x-4}{6}= \frac{3x-12}{18}\)

Step 3: Combine fractions

Now we can write the inequality as a single fraction, and simplify -> \(\frac{3x-12}{18} \geq \frac{2x+3}{18}\)

Step 4: Remove the denominator

Since the denominators on both sides of the inequality are equal, we can eliminate them -> \(3x -12 \geq 2x + 3\)

Step 5: Solve the inequality

Finally, we can solve for x. Subtract 2x from both sides to get -> \(x-12 \geq 3\). Then, add 12 to both sides to get -> \(x \geq 15\)

Step 6: Graph the solution

This inequality includes the value 15, so on a number line, we will include 15 in the solution set. Therefore, we would draw a closed dot on 15 and shade the area to the right of it since x is greater than or equal to 15.

Key Concepts

Interval Notation

Interval notation is a way to express solution sets of inequalities. Instead of using phrases like "greater than" or "less than," we use symbols and brackets to show where the values lie.

For example, if an inequality is solved with a solution like "\(x \geq 15\)," it means the solution includes 15 and all numbers greater than 15. In interval notation, this is written as "[15, \infty)."

• The square bracket "[" means the number is included in the set.
• The parenthesis ")" indicates that infinity is not a specific number and is not included.

Interval notation effectively communicates the range of possible solutions in a concise manner.

Solution Set

A solution set is a collection of all possible values that satisfy an inequality. When you solve a linear inequality, like \(x \geq 15\), the solution set contains every number that works for the inequality.

This encompasses both whole numbers and decimals. Understanding the solution set helps you see all possible answers to the inequality.

• Based on the inequality's direction, the solution set can include numbers above, below, or equal to a specific value.
• In our example, the solution set is unbounded to the right, as it includes all numbers greater than or equal to 15.

Finding a solution set is crucial as it represents all the possibilities for the variable in question.

Graphical Solution

A graphical solution represents the solution set visually on a number line. It provides a clear picture of where the solutions are located.

To graph \(x \geq 15\) on a number line, you would:

• Draw a closed dot on 15, indicating that 15 is included in the solution.
• Shade the line to the right of 15, showing all numbers greater than 15 are included in the solution.

This visual representation helps solidify understanding by giving a clear image of the inequality's solution set.

Number Line

The number line is a straight line with numbers placed at equal intervals along its length. It's a valuable tool for representing numbers and inequalities.

On the number line, each point corresponds to a number. When graphing solutions, the number line helps show where specific values and ranges live.

• Closed dots represent numbers that are included in the solution set.
• Open dots, used in other inequalities, represent numbers not included.

By shading parts of the number line, you visualize where the solution set exists, making inequalities easier to understand and interpret.