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Chapter 2: Problem 38

Solve. Write the solution set using interval notation. See Examples 1 through 7. $$ -x>-2 $$

Short Answer

Expert verified
The solution set is \((-\infty, 2)\).

Step by step solution

01

Understand the inequality

We start with the inequality \(-x > -2\). Our goal is to solve for \(x\) to find the solution set.
02

Multiply by -1

To remove the negative sign in front of \(x\), we multiply both sides of the inequality by \(-1\). Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. Thus, \(-x > -2\) becomes \(x < 2\).
03

Write the solution in interval notation

The inequality \(x < 2\) can be expressed in interval notation as \((-\infty, 2)\), which represents all real numbers less than 2.

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Key Concepts

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

Inequality Notation
Inequality notation is a way of expressing mathematical sentences that state that two values are not equal, but there is a specific relationship between them, often involving less than or greater than. In inequalities, we use symbols like:
  • <: less than
  • >: greater than
  • 鈮�: less than or equal to
  • 鈮�: greater than or equal to
These symbols help describe the relationship between two expressions, similar to equations, but without stating equality. For example, \(a < b\) means that a is less than b. This notation is crucial in algebra for defining ranges of possible solutions.
When solving inequalities, like \(-x > -2\), our aim is to isolate the variable, similar to solving equations.
Interval Notation
Interval notation is a compact way to describe a range of numbers on the number line. It uses parentheses and brackets to show where the range starts and ends, and whether those endpoints are included.
  • Parentheses ( ) denote that an endpoint is not included, meaning the range is open.
  • Brackets [ ] signify that an endpoint is included, indicating a closed range.
For the inequality \(x < 2\), the interval notation is written as \((-\infty, 2)\). This means all numbers less than 2, without including 2 itself. The use of \(-\infty\) indicates that there is no lower bound to the range, as infinity isn't an actual number, but a concept showing the range continues indefinitely.
Reversing Inequality Signs
Reversing inequality signs is an important rule to remember when solving inequalities, especially when multiplying or dividing both sides by a negative number. When you do this, you must flip the inequality sign to maintain the correct relationship between the expressions. For example:
  • From \(-x > -2\)\,, multiplying both sides by -1 changes it to \(x < 2\).
This rule is crucial because it ensures that the inequality remains valid under the transformation.
Neglecting to reverse the sign would lead to incorrect results, misleading the solution's interpretation. Therefore, whenever you perform operations involving a negative multiplier or divisor, remember to flip the inequality sign to keep solutions accurate and meaningful.

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Most popular questions from this chapter

The expression \(\left|x_{T}-x\right|\) is defined to be the absolute error in \(x\) where \(x_{T}\) is the true value of a quantity and \(x\) is the measured value or value as stored in a computer. If the true value of a quantity is 0.2 and the approximate value stored in a computer is \(\frac{51}{256},\) find the absolute error.

The calorie count of a serving of food can be computed based on its composition of carbohydrate, fat, and protein. The calorie count \(C\) for a serving of food can be computed using the formula \(C=4 h+9 f+4 p,\) where \(h\) is the number of grams of carbohydrate contained in the serving, \(f\) is the number of grams of fat contained in the serving, and p is the number of grams of protein contained in the serving. Solve this formula for \(f,\) the number of grams of fat contained in a serving of food.

Describe how solving \(|x+4|=0\) is similar to solving \(|x+4| \leq 0\)

Solve each inequality. Graph the solution set and write it in interval notation. $$ \left|\frac{7+x}{2}\right| \geq 4 $$

Solve each inequality. Graph the solution set and write it in interval notation. $$ |-3+x| \leq 10 $$
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