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Chapter 4: Problem 38

Solve. Then graph. Write the solution set using both set-builder notation and interval notation. $$ 2 x-5 \geq 9 $$

Short Answer

Expert verified
Set-builder: \( \{ x | x \geq 7 \} \), Interval: \( [7, \infty) \)

Step by step solution

01

- Isolate the variable

Start by isolating the variable on one side of the inequality. Add 5 to both sides of the inequality to get:\[ 2x - 5 + 5 \geq 9 + 5 \]This simplifies to:\[ 2x \geq 14 \]
02

- Solve for the variable

Next, solve for the variable by dividing both sides of the inequality by 2. This gives:\[ \frac{2x}{2} \geq \frac{14}{2} \]Simplifying this results in:\[ x \geq 7 \]
03

- Write the solution set

Now, express the solution set in both set-builder notation and interval notation:Set-builder notation: \[ \{ x | x \geq 7 \} \]Interval notation: \[ [7, \infty) \]
04

- Graph the solution

To graph the solution, plot a number line and shade the region starting from 7 and extending to positive infinity. Use a closed circle at 7 to indicate that 7 is included in the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set-Builder Notation
Set-builder notation is a concise way to express a set of numbers that satisfy a certain condition. It is particularly useful when dealing with inequalities. In this notation, we describe the set in terms of the variable and the condition it must meet. For example, the solution \( x \geq 7 \) from our exercise can be written in set-builder notation as follows:
\[ \{ x \mid x \geq 7 \} \] This reads as 'the set of all x such that x is greater than or equal to 7'. It's a standard way to succinctly describe an infinite set.
Interval Notation
Interval notation is another method to represent sets of numbers, especially when dealing with ranges and inequalities. This notation indicates the start and end values of the interval, including whether these boundaries are part of the set. Using parentheses \( ( ) \) denotes that the boundary isn't included, while brackets \( [ ] \) mean the boundary is included.
In our exercise, the inequality \( x \geq 7 \) translates to interval notation as:
\[ [7, \infty) \] This means the set includes all numbers from 7 to positive infinity, including 7 but not infinity, since infinity is not a specific number and cannot be included.
Graphing Inequalities
Graphing inequalities helps visualize the solutions on a number line, making it easier to see which numbers are included. Here鈥檚 how to graph the solution \( x \geq 7 \):
  • Draw a number line.
  • Locate the number 7 on the number line.
  • Place a closed circle at 7 to indicate that 7 is included in the solution.
  • Shade the region to the right of 7, extending to positive infinity. This shaded part represents all the numbers greater than or equal to 7.
The closed circle on 7 shows that 7 is part of the solution, while the shaded area indicates all the numbers that satisfy the inequality.

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Most popular questions from this chapter

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ 6+|3-2 x|>10 $$

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