In This Discussion You Will Be Demonstrating Your Understand

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. View the example attached. Your “and” compound inequality is: —1

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Understanding compound inequalities is fundamental in algebra because they allow us to describe ranges of solutions that satisfy multiple conditions simultaneously. The problem considers both an “and” (conjunctive) compound inequality and an “or” (disjunctive) compound inequality, illustrating the different solution strategies and implications of each.

The “and” compound inequality given is: —1

—1 - 4

which simplifies to:

—5

Next, I recognize that dividing by a negative number will flip the inequality signs. Since the coefficient of x is —2, dividing through by —2 requires reversing the inequalities:

\[ \frac{—5}{—2} > x \geq \frac{1}{—2} \]

which simplifies to:

\[ \frac{5}{2} > x \geq -\frac{1}{2} \]

This inequality can be rewritten in a more standard interval notation as:

\[ -\frac{1}{2} \leq x

This solution set indicates that for the combined condition to be true, x must be greater than or equal to —1/2 and less than 5/2. Graphically, this can be represented as a line segment on the number line starting at —1/2 (closed circle) and ending at 5/2 (open circle), with all points in between included in the solution set, illustrating the intersection of the half-open interval.

Now, considering the “or” compound inequality: x

3x ≥ 6

x ≥ 2

Thus, the solution set for the “or” inequality is:

\[ (-\infty, 0) \cup [2, \infty) \]

This union of intervals includes all x less than zero and all x greater than or equal to 2, indicating that either condition being true satisfies the compound inequality. Graphically, this is depicted as two separate line segments: one extending infinitely to the left up to 0 (but not including 0) and another starting at 2 (including 2) and extending to infinity.

Interpretation of these solution sets in words reveals that the “and” inequality restricts x to a specific interval where both conditions hold simultaneously—bounded between —1/2 and 5/2—symbolizing a range of values satisfying the conjunction. In contrast, the “or” inequality illustrates a union of two disjoint intervals, representing all x values less than 0 or at least 2, with no solutions between 0 and 2. These differences underscore the significant impact whether inequalities are combined through “and” or “or.”

Using the five vocabulary words, we see that in the case of the “and” compound inequality, the solution is the intersection of the two individual solutions, which narrows the possible x-values. Conversely, in the “or” compound inequality, the solution comprises the union of the two solution sets, broadening the range of solutions. These concepts are critical in understanding how compound inequalities function and influence the set of solutions in algebraic problems.

In conclusion, solving compound inequalities requires careful attention to operations such as dividing by negative numbers, which necessitates reversing inequalities. Recognizing the difference between intersection and union helps in accurately portraying the solution sets both algebraically and graphically. These tools are essential for solving real-world problems where multiple conditions must be satisfied simultaneously or alternatively, thus demonstrating the importance of compound inequalities in mathematical reasoning and application.

References

  • Blitzer, R. (2017). Algebra and Trigonometry. Pearson.
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