Keyarra And Jarome Were Simplifying The Inequality $2x - 2 \ \textless \ 3x - 7$. When Comparing Their Work, They Discovered That They Solved The Problem Differently. Is Both Of Their Work Correct? Justify Your Thinking.*Hint: If Their

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Simplifying Inequalities: A Comparative Analysis of Keyarra and Jarome's Work

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. When simplifying inequalities, it is essential to follow the correct procedures to ensure that the solution is accurate. In this article, we will explore the work of Keyarra and Jarome, two students who attempted to simplify the inequality 2x−2 \textless 3x−72x - 2 \ \textless \ 3x - 7. We will analyze their work, identify any errors, and provide a justification for our findings.

Keyarra's approach to simplifying the inequality is as follows:

  1. Add 2 to both sides: Keyarra starts by adding 2 to both sides of the inequality, resulting in 2x \textless 3x−52x \ \textless \ 3x - 5.
  2. Subtract 3x from both sides: Next, Keyarra subtracts 3x from both sides, yielding −x \textless −5-x \ \textless \ -5.
  3. Multiply both sides by -1: Finally, Keyarra multiplies both sides by -1, obtaining x \textgreater 5x \ \textgreater \ 5.

Jarome's approach to simplifying the inequality is as follows:

  1. Add 2 to both sides: Jarome starts by adding 2 to both sides of the inequality, resulting in 2x \textless 3x−52x \ \textless \ 3x - 5.
  2. Subtract 2x from both sides: Next, Jarome subtracts 2x from both sides, yielding 0 \textless x−50 \ \textless \ x - 5.
  3. Add 5 to both sides: Finally, Jarome adds 5 to both sides, obtaining 5 \textless x5 \ \textless \ x.

At first glance, both Keyarra and Jarome's approaches seem to be correct. However, upon closer inspection, we can identify some errors in their work.

Keyarra's Error

Keyarra's mistake lies in the third step, where she multiplies both sides by -1. When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. In this case, Keyarra should have obtained x \textless 5x \ \textless \ 5 instead of x \textgreater 5x \ \textgreater \ 5.

Jarome's Error

Jarome's mistake lies in the second step, where he subtracts 2x from both sides. This step is unnecessary and can be avoided by adding 2 to both sides of the original inequality. Additionally, Jarome's final step of adding 5 to both sides is correct, but the direction of the inequality sign should be reversed when multiplying or dividing both sides by a negative number.

In conclusion, while both Keyarra and Jarome's approaches to simplifying the inequality 2x−2 \textless 3x−72x - 2 \ \textless \ 3x - 7 seem to be correct at first glance, they both contain errors. Keyarra's mistake lies in multiplying both sides by -1, while Jarome's mistake lies in subtracting 2x from both sides and not reversing the direction of the inequality sign when multiplying or dividing both sides by a negative number. By carefully analyzing their work and identifying the errors, we can provide a justification for our findings and ensure that the solution is accurate.

When simplifying inequalities, it is essential to follow the correct procedures to ensure that the solution is accurate. Here are some tips to keep in mind:

  • Add or subtract the same value to both sides: When adding or subtracting the same value to both sides of an inequality, the direction of the inequality sign remains the same.
  • Multiply or divide both sides by a positive number: When multiplying or dividing both sides of an inequality by a positive number, the direction of the inequality sign remains the same.
  • Multiply or divide both sides by a negative number: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.
  • Avoid unnecessary steps: When simplifying inequalities, it is essential to avoid unnecessary steps that can lead to errors.

By following these tips and carefully analyzing the work of Keyarra and Jarome, we can provide a justification for our findings and ensure that the solution is accurate.
Frequently Asked Questions: Simplifying Inequalities

A: An inequality is a mathematical statement that compares two expressions using a symbol such as <, >, ≤, or ≥. Inequalities are used to describe relationships between variables and can be used to solve a wide range of problems.

A: The basic rules for simplifying inequalities are:

  • Add or subtract the same value to both sides: When adding or subtracting the same value to both sides of an inequality, the direction of the inequality sign remains the same.
  • Multiply or divide both sides by a positive number: When multiplying or dividing both sides of an inequality by a positive number, the direction of the inequality sign remains the same.
  • Multiply or divide both sides by a negative number: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.
  • Avoid unnecessary steps: When simplifying inequalities, it is essential to avoid unnecessary steps that can lead to errors.

A: To simplify an inequality with fractions, follow these steps:

  1. Multiply both sides by the least common multiple (LCM) of the denominators: This will eliminate the fractions and make it easier to simplify the inequality.
  2. Simplify the inequality: Once the fractions have been eliminated, simplify the inequality by adding or subtracting the same value to both sides, or multiplying or dividing both sides by a positive or negative number.
  3. Reverse the direction of the inequality sign if necessary: If you multiplied both sides by a negative number, reverse the direction of the inequality sign.

A: To simplify an inequality with absolute values, follow these steps:

  1. Write the inequality as a double inequality: An absolute value inequality can be written as a double inequality, where the expression inside the absolute value is greater than or equal to a certain value, and less than or equal to a certain value.
  2. Simplify the inequality: Once the absolute value has been eliminated, simplify the inequality by adding or subtracting the same value to both sides, or multiplying or dividing both sides by a positive or negative number.
  3. Reverse the direction of the inequality sign if necessary: If you multiplied both sides by a negative number, reverse the direction of the inequality sign.

A: Some common mistakes to avoid when simplifying inequalities include:

  • Not reversing the direction of the inequality sign when multiplying or dividing both sides by a negative number
  • Not adding or subtracting the same value to both sides
  • Not multiplying or dividing both sides by a positive or negative number
  • Not avoiding unnecessary steps

A: To practice simplifying inequalities, try the following:

  • Work through examples: Practice simplifying inequalities by working through examples and exercises.
  • Use online resources: There are many online resources available that can help you practice simplifying inequalities, such as video tutorials and practice problems.
  • Ask for help: If you are struggling to simplify an inequality, ask for help from a teacher, tutor, or classmate.

By following these tips and practicing simplifying inequalities, you can become more confident and proficient in solving inequalities.