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Chapter 1: Problem 10

\(\quad\) A student solved the following inequality incorrectly as shown. $$ \begin{array}{l} -2 x<-18 \\ \frac{-2 x}{-2}<\frac{-18}{-2} \end{array} $$ $$ x<9 \quad(-\infty, 9) $$ WHAT WENT WRONG? Give the correct solution set.

Short Answer

Expert verified
The student forgot to reverse the inequality when dividing by a negative number. The correct solution is \( x > 9 \) or \( (9, \infty) \).

Step by step solution

01

- Identify the Original Inequality

The given inequality is \( -2x < -18 \).
02

- Remember the Rule for Dividing by a Negative Number

When dividing both sides of an inequality by a negative number, the direction of the inequality must be reversed.
03

- Correctly Divide Both Sides by -2

Divide both sides of the inequality by -2 and reverse the inequality sign: \( \frac{-2x}{-2} > \frac{-18}{-2} \). This simplifies to \( x > 9 \).
04

- Generate the Correct Solution Set

The correct solution set is \( (9, \infty) \).

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Key Concepts

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

inequality rules
Inequalities are expressions involving the symbols <, >, 鈮�, and 鈮�. Understanding how they work is crucial in solving them correctly.
  • Basic Inequality Rules: You can add, subtract, multiply, and divide both sides by the same positive number without changing the direction of the inequality.
  • Reversing the Inequality: When multiplying or dividing by a negative number, you must reverse the inequality sign. This rule often trips up students, so always pay extra attention!
  • Comparing Expressions: Always ensure your final answer matches the context of the problem.
In this exercise, the student forgot to reverse the inequality sign when dividing by a negative number.
Following these rules will help you avoid similar mistakes.
dividing by negative numbers
Dividing by a negative number isn't just a simple arithmetic operation in the context of inequalities; it also changes the direction of the inequality.
  • Key Concept: While with equations, dividing by a negative number does not affect the outcome beyond the numerical value; with inequalities, this changes the entire inequality.
  • Practical Example: Starting with the inequality \-2x < \-18, divide both sides by \-2, remembering to flip the inequality sign: \ \( \frac{-2x}{-2} > \frac{-18}{-2} \) \ Simplified, we get \( x > 9 \).
  • Avoiding Errors: Keep the rule in mind at all times to avoid errors like in the provided problem.
Always ensure to perform an inequality check step after solving.
For instance, by substituting values back into the original inequality to verify correctness.
solution set
The solution set of an inequality represents all the possible values that satisfy the inequality.
  • Expression Representation: Inequalities use intervals to represent solution sets. For example, \( x > 9 \) is represented as \( (9, \infty) \).
  • Interpreting Intervals: - When the inequality is strict (< or >), use parentheses '('. When it's inclusive (鈮� or 鈮�), use brackets '['.
  • Correct Solution: For our solved problem, the solution is \( x > 9 \). Thus, our solution set is \( (9, \infty) \).
Clearly stating the solution set helps in understanding the range of valid values.
Always double-check that your intervals and signs are correct to avoid confusion.

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Most popular questions from this chapter

Solve each equation or inequality. $$ |0.5 x-3.5|+0.2 \geq 0.6 $$

The 10 tallest buildings in Houston, Texas, are listed along with their heights. $$ \begin{array}{|l|c|} \hline \quad {\text { Building }} & \text { Height (in feet) } \\ \hline \text { JPMorgan Chase Tower } & 1002 \\ \text { Wells Fargo Plaza } & 992 \\ \text { Williams Tower } & 901 \\ \text { Bank of America Center } & 780 \\ \text { Texaco Heritage Plaza } & 762 \\ \text { 609 Main at Texas } & 757 \\ \text { Enterprise Plaza } & 756 \\ \text { Centerpoint Energy Plaza } & 741 \\ \text { 1600 Smith St. } & 732 \\ \text { Fulbright Tower } & 725 \\ \hline \end{array} $$ Use this information. Let \(k\) represent the average height of these buildings. If a height \(x\) satisfies the inequality $$ |x-k|

Solve each equation or inequality. $$ |0.5 x-3.5|+0.2 \geq 0.6 $$

Give, in interval notation, the unknown numbers in each description. Three times a number, minus \(5,\) is no more than 7 .

The 10 tallest buildings in Houston, Texas, are listed along with their heights. $$ \begin{array}{|l|c|} \hline \quad {\text { Building }} & \text { Height (in feet) } \\ \hline \text { JPMorgan Chase Tower } & 1002 \\ \text { Wells Fargo Plaza } & 992 \\ \text { Williams Tower } & 901 \\ \text { Bank of America Center } & 780 \\ \text { Texaco Heritage Plaza } & 762 \\ \text { 609 Main at Texas } & 757 \\ \text { Enterprise Plaza } & 756 \\ \text { Centerpoint Energy Plaza } & 741 \\ \text { 1600 Smith St. } & 732 \\ \text { Fulbright Tower } & 725 \\ \hline \end{array} $$ Use this information. Work each of the following. (a) Write an absolute value inequality that describes the height of a building that is not within \(95 \mathrm{ft}\) of the average. (b) Solve the inequality from part (a). (c) Use the result of part (b) to list the buildings that are not within \(95 \mathrm{ft}\) of the average. (d) Confirm that the answer to part (c) makes sense by comparing it with the answer to Exercise 131 .
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