In This Discussion You Will Be Demonstrating Your Und 122966
In This Discussion You Will Be Demonstrating Your Understanding Of Co
In this discussion, you will be demonstrating your understanding of compound inequalities and the effect that dividing by a negative has on an inequality. Read the following instructions in order and view the example (available for download in your online classroom) to complete this discussion. Please complete the following problems according to your assigned number. (Instructors will assign each student their number.) Your “and” compound inequality is: –2 0 or – 2x ≥ 4. Solve the compound inequalities as demonstrated in Elementary and Intermediate Algebra and the Instructor Guidance in the left navigation toolbar, in your online course.
Be careful of how a negative x-term is handled in the solving process. Show all math work arriving at the solutions. Show the solution sets written algebraically and as a union or intersection of intervals. Describe in words what the solution sets mean, and then display a simple line graph for each solution set. This is demonstrated in the Instructor Guidance in the left navigation toolbar, in your online course.
Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in sentences describing your math work. Compound inequalities, And, Or, Intersection, Union. Your initial post should be at least 250 words in length. Support your claims with examples from required material(s) and/or other scholarly resources, and properly cite any references. Respond to at least two of your classmates’ posts by Day 7.
Paper For Above instruction
Understanding and solving inequalities constitute fundamental skills in algebra, crucial for analyzing relationships between variables. Specifically, compound inequalities involve two or more inequalities combined using logical connectors such as and or or. These distinctions significantly influence the solution approaches and the interpretation of the solution sets. In this discussion, I focus on two problems: an and compound inequality and an or compound inequality, illustrating their solutions, representations, and interpretations.
Solution of the And Compound Inequality
The inequality –2
–2 – 26
which simplifies to: –28
Next, dividing all parts of the inequality by 7 — and here, I need to be cautious because dividing by a positive number does not change the direction of the inequality signs:
(–28)/7
which simplifies to: –4
Written in interval notation, the solution set is:
(–4, –22/7].
This solution set represents all real numbers x that are greater than –4 but less than or equal to –22/7. In words, the solution includes all values between –4 and approximately –3.14, including –3.14 itself but not –4.
Graphically, this is represented by a line segment starting just to the right of –4 and extending to –22/7, with a closed circle at –22/7 and an open circle at –4.
Solution of the Or Compound Inequality
The second inequality, 3x + 8 > 0 or – 2x ≥ 4, involves an or conjunction. I solve each inequality separately:
For 3x + 8 > 0:
3x > –8
x > –8/3
For – 2x ≥ 4:
Dividing both sides by –2 (and reversing the inequality sign because dividing by a negative):
x ≤ –2
Therefore, the solution for the first inequality is x > –8/3, and the second is x ≤ –2.
Since these inequalities are connected by or, the combined solution is the union of the individual solutions: all x > –8/3 or x ≤ –2. In interval notation, this is:
(–∞, –2] ∪ (–8/3, ∞)
The interpretation of this solution is that any x less than or equal to –2, or any x greater than –8/3, satisfies at least one of the inequalities. Graphically, this is represented by two rays: one extending leftward from –2, including –2, and the other extending rightward from –8/3, not including –8/3.
Final Remarks
Understanding the difference between an and and an or compound inequality is key in solving inequalities accurately. The and inequality's solution is the intersection of the individual solutions, meaning the solutions common to both inequalities. Conversely, the or inequality's solution is the union of the solution sets, encompassing all values satisfying either inequality. The process of solving these inequalities emphasizes the importance of carefully handling negative coefficients, especially when dividing, as this operation can reverse the inequality's direction. Visual representations, such as line graphs, reinforce an understanding of the solution sets' nature and extent.
References
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