Describe In Words What The Solution Sets Mean And Then Displ
Describe In Words What The Solution Sets Mean And Then Display A Simp
Describe in words what the solution sets mean, and then display a simple line graph for each solution set as demonstrated in the Instructor Guidance in the left navigation toolbar. 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. The problem is - your "and" compound inequality is "-10 "or" compound inequality is "4x + 7
Paper For Above instruction
The task involves interpreting and visualizing the solution sets of two compound inequalities. First, I will analyze the “and” compound inequality: -10 intersection of their solution sets, meaning we are interested in the values of x that satisfy both inequalities simultaneously. The process to solve this involves isolating x across the inequality chain.
Starting with -10 "and" inequality is all x values strictly between -1 and 3. Conceptually, this represents the intersection of the two individual solutions: x-values greater than -1 and less than 3.
Next, I examine the "or" compound inequality: 4x + 7 "or" indicates that we are looking at the union of the two solution sets, meaning any x that satisfies either of the inequalities.
Starting with the first part: 4x + 7
The second part: 1 – x ≤ -2, subtract 1 from both sides to get -x ≤ -3. Multiply both sides by -1 (remembering to reverse the inequality sign when multiplying or dividing by a negative number), resulting in x ≥ 3.
The union of these solution sets is all x-values less than 1, combined with all x-values greater than or equal to 3. The total solution set for the "or" inequality includes the two disjoint intervals: x
Visualizing these solutions on a number line helps clarify their meaning. The intersection of the first inequality is represented by a segment between -1 and 3, both endpoints excluded, indicating x-values that satisfy both conditions simultaneously. The union of the second inequality covers two regions: all points less than 1 and all points greater than or equal to 3, with a gap in between where x values do not satisfy either condition.
In conclusion, the "and" compound inequality produces a bounded solution set, indicating the set of x-values where multiple conditions overlap. Conversely, the "or" compound inequality captures an unbounded solution set made up of two separate parts, emphasizing the importance of understanding union in combining solution sets.
Graphically, the first solution set is a line segment between -1 and 3, and the second represents two distinct rays: one extending to the left from negative infinity to 1, and the other extending from 3 to positive infinity. These visualizations are vital in understanding how these inequalities define ranges of solutions and their relationships.
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