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Chapter 15: Problem 49

Solve and graph. Write each answer in set-builder notation and in interval notation. $$ 2 x \geq-3 $$

Short Answer

Expert verified
Set-builder notation: \[ \{ x | x \geq -\frac{3}{2} \} \]. Interval notation: \[ \left[ -\frac{3}{2}, \infty \right) \].

Step by step solution

01

- Isolate the variable

To isolate the variable, divide both sides of the inequality by 2. \[ \frac{2x}{2} \geq \frac{-3}{2} \] Simplifying this, we get \[ x \geq -\frac{3}{2} \]
02

- Write the solution in set-builder notation

In set-builder notation, the solution is written as: \[ \{ x | x \geq -\frac{3}{2} \} \]
03

- Write the solution in interval notation

In interval notation, the solution is written as: \[ \left[ -\frac{3}{2}, \infty \right) \]
04

- Graph the solution

To graph the solution, draw a number line and shade the region starting from \[ -\frac{3}{2} \] and extending to infinity. Use a closed circle at \[ -\frac{3}{2} \] to indicate that \[ x = -\frac{3}{2} \] 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 mathematical way of describing a set by stating the properties that its members must satisfy. Instead of listing every element in the set, we provide a rule or a condition. For example, the solution to \(2x \geq-3\) was written as \(\{ x | x \geq -\frac{3}{2} \}\). The vertical bar \('|\) can be read as 'such that' and the expression states that we're considering all \(x\) such that \(x\) is greater than or equal to \(-\frac{3}{2}\). This is a compact and precise way to represent the solution.
interval notation
Interval notation is a way of representing a set of numbers along a number line. It uses parentheses \(()\) and brackets \([]\) to describe intervals. For the inequality \(2x \geq -3\), after isolating \(x\), we write the solution \(x \geq -\frac{3}{2}\) in interval notation as \([-\frac{3}{2}, \infty)\).
Here, the square bracket \([\) next to \(-\frac{3}{2}\) means that \(-\frac{3}{2}\) is included in the set, while the parenthesis \(()\) next to infinity indicates that infinity is not a specific value we reach or include but more of a direction.
graphing solutions
Graphing solutions on a number line gives a visual representation of the set of possible values. For \(x \geq -\frac{3}{2}\), follow these steps:
  • Draw a horizontal number line.
  • Locate \(-\frac{3}{2}\) on the number line and mark it with a closed circle to show that this value is part of the solution. A closed circle indicates that the endpoint is included.
  • Shade the region to the right of \(-\frac{3}{2}\) towards positive infinity. This shading signifies that all numbers greater than or equal to \(-\frac{3}{2}\) are in the solution set.
This visual aid helps to understand the scope of the solution set easily.

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