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Chapter 2: Problem 12

Graph the solutions of each inequality on a number line. \(-2 \leq x \leq 0\)

Short Answer

Expert verified
The number line will have a closed circle at -2 and 0, with a line segment drawn between both points, representing the solutions of the inequality.

Step by step solution

01

- Understanding the inequality and plotting it on the number line

In the inequality \(-2 \leq x \leq 0\), \(x\) is greater than or equal to -2 and less than or equal to 0. Start by drawing a number line. Then, mark the numbers -2 and 0 distinctly on this number line. Because these inequalities include an equality condition, the point for -2 and 0 on the number line should be a closed circle, indicating that these numbers are included in the solution.
02

- Highlighting the solution set

Now, all values that lie between -2 and 0 inclusive need to be represented. Draw a line segment between the marked points of -2 and 0 to represent all numbers that exist between them. This line is the graphical representation of the solutions.

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Key Concepts

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

Number Line
A number line is a visual representation of numbers placed at equal intervals along a straight line. It helps in understanding and performing operations involving integers and real numbers. To graph inequalities like \[-2 \leq x \leq 0\]on a number line, you need to follow these steps:
  • First, draw a horizontal line, which will serve as your number line.
  • Mark relevant points, here \(-2\) and \(0\), prominently.
  • Use closed circles on these numbers, as the inequality signs \(\leq\) mean that \(-2\) and \(0\) are included in the solution set.
  • Shade or draw a line segment between these two points to show that all numbers between them are included.

This simple depiction makes identifying possible solutions and understanding integer relationships straightforward.
Inequality Solutions
Understanding inequality solutions is crucial for solving algebraic problems. Inequalities express a range of possible solutions, rather than just a single one. They use different symbols:
  • \(<\) or \(>\): indicates values that are not included, represented with open circles on a number line.
  • \(\leq\) or \(\geq\): signifies inclusive values, represented with closed circles.

For the given inequality \[-2 \leq x \leq 0\], the solution involves all values between \(-2\) and \(0\), including the endpoints. This means any number like \(-1.5\) or \(0\) satisfies the inequality. These solutions are depicted as a continuous section on the number line. It's essential to differentiate when to use open or closed circles based on whether values are inclusive or exclusive in a solution.
Algebra
Algebra forms the backbone of many mathematical concepts including solving inequalities. In algebra, inequalities are dealt with using similar techniques as equations, but with extra rules.
  • When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.
  • Addition or subtraction keeps the inequality sign intact, just as when solving equations.

For instance, in our exercise with \(-2 \leq x \leq 0\), the algebraic representation involves ensuring solutions remain valid throughout operations. With inequalities, knowing how to manipulate and graph the solution set enhances understanding of variable ranges and possible real-world applications. These algebraic principles enable one to solve practical problems involving constraints and limits.

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Most popular questions from this chapter

After a \(30 \%\) reduction, you purchase a DVD player for \(\$ 98 .\) What was the price before the reduction?

In a film, the actor Charles Coburn played an elderly "uncle" character criticized for marrying a woman when he is 3 times her age. He wittily replies, "Ah, but in 20 years time I shall only be twice her age." How old is the "uncle" and the woman?

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. Nine times a number is 30 more than three times that number. Find the number.

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. A number increased by 60 is equal to \(410 .\) Find the number.

Will help you prepare for the material covered in the next section. Is 6 a solution of \(4 y-7 \geq 5 ?\)
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