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Chapter 2: Problem 22

Write the solution set in interval notation and graph it. \(x \geq-7\)

Short Answer

Expert verified
The solution in interval notation is \[ [-7, +\infty) \] and can be graphed with a filled circle at \[ -7 \] and a shaded line extending to the right.

Step by step solution

01

- Identify the inequality

The given inequality is written as: \[ x \geq -7 \]
02

- Convert to interval notation

In interval notation, this inequality includes all numbers from \[ -7 \] to \[ + \infty \]. Since \[ -7 \] is included in the solution set, we use a square bracket. The interval notation is: \[ [-7, +\infty) \]
03

- Graph the solution

To graph the solution: 1. Draw a number line. 2. Place a filled-in circle at \[ -7 \] (indicating that \[ -7 \] is included in the solution). 3. Shade the number line to the right of \[ -7 \] (indicating all numbers greater than or equal to \[ -7 \] are part of the solution).

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Key Concepts

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

interval notation
Interval notation is a way of writing subsets of the real number line. It uses brackets to show where the intervals start and end. For example, the interval \([-7, +\infty)\) represents all numbers between -7 and infinity, including -7 but not including infinity. The square bracket \([\) means the number is included in the interval, while the parenthesis \(\) means the number is not included. In our case, \([-7, +\infty)\), -7 is included, so we use a square bracket. Infinity does not end, so we use a parenthesis. For inequalities, interval notation is a compact way to describe all solutions of the given inequality.
number line graph
A number line graph visually represents solutions to an inequality. To create a number line graph for the inequality \[ x \geq -7 \], you start by drawing a horizontal line. Label a point on the line as -7. Since -7 is included in the solution set, we place a filled-in circle on -7. Next, shade the line to the right of -7 to indicate all numbers greater than or equal to -7 are part of the solution. This visual representation helps understand the range of solutions quickly.
solution set
The solution set of an inequality is the set of all values that make the inequality true. For the inequality \[ x \geq -7 \], the solution set includes all real numbers starting from -7 to infinity. Writing the solution in interval notation as \[ [-7, +\bollinfty) \] clearly shows the range of values. And graphing it on a number line confirms the visual span of the solutions. Both methods ensure that anyone can easily understand and verify all possible values that satisfy the inequality.

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Most popular questions from this chapter

Discussion. For which of the inequalities is the notation used correctly? a) \(-2 \leq x<3\) b) \(-4 \geq x<7\) c) \(-1 \leq x>0\) d) \(6

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