Problem 107

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) -x^{2}-x \leq 0$$ $$(b) -x^{2}-x>0$$

Step-by-Step Solution

Verified

Answer

(a) $[-1, 0]$ \cup (0, \infty)\), (b) $(-1, 0)$.

1Step 1: Understand the inequality

Consider the inequality $-x^2 - x \leq 0$.Rearrange the inequality to see it clearly as $-x(x + 1) \leq 0$.This signifies that $x(x + 1) \geq 0$.

2Step 2: Find the roots

To solve $x(x + 1) = 0$,find the roots; the roots of the equation are $x = 0$ and $x = -1$.

3Step 3: Test intervals

Consider the intervals defined by the roots: $(-\infty, -1)$, $(-1, 0)$, $(0, \infty)$. Test a point in each interval to determine where the inequality $x(x + 1) \geq 0$ holds.

4Step 4: Evaluate intervals

- Choose $x = -2$ in interval $(-\infty, -1)$: $(-2)(-2 + 1) = 2$, positive.- Choose $x = -0.5$ in interval $(-1, 0)$: $(-0.5)(-0.5 + 1) = -0.25$, negative.- Choose $x = 1$ in interval $(0, \infty)$: $(1)(1 + 1) = 2$, positive.So the inequality holds for $(-\infty, -1]$ and $[0, \infty)$.

5Step 5: Solve Part (b)

Now for $-x^2 - x > 0$,convert it to $x(x + 1) < 0$.From the previous step, we determined this condition is satisfied in the interval $(-1, 0)$.

6Step 6: Graphical Support

Graph $y = -x^2 - x$. Identify the intervals where $y \leq 0$ and $y > 0$. Confirm graphically that the intervals $[-1, 0)$ and ((-1, 0)) fit the conditions for the inequalities.

Key Concepts

Analytical SolutionsGraphical SupportRoots of EquationsInterval Testing

Analytical Solutions

In mathematics, solving inequalities analytically involves manipulating the inequality through algebraic expressions to find solutions. To begin this process, it is essential to understand the given inequality clearly. Let's use the inequality \[-x^2 - x \leq 0\]We first rearrange it to simplify the notation and understanding. By factoring, we transform it into \[-x(x+1) \leq 0\]This rearrangement helps us to see that our task is to determine when the expression on the left is less than or equal to zero. This transformation is crucial as it lays the foundation for identifying critical points, or the roots.

Graphical Support

Visualizing inequalities helps to solidify the understanding of the solution set. To support our analytical findings for the inequality \[-x^2 - x \leq 0\]we can graph the equation:\[y = -x^2 - x\]The graph of this quadratic function is a downward-opening parabola. Key points where this curve crosses the x-axis are the roots of the quadratic, specifically at \[x = 0\] and \[x = -1\].The graph will visually confirm where the parabola is below or touches the x-axis, clearly showing regions where the inequality \[-x^2 - x \leq 0\] holds, meaning the area where the graph is at or below the x-axis. This gives us insight into the intervals where our solutions lie, backing up the analytical approach.

Roots of Equations

To solve inequalities like \[-x^2 - x \leq 0\] finding the roots of the corresponding equation \[-x(x+1) = 0\] is a fundamental step. The roots here are determined by setting each factor in the equation to zero:

Setting \[x = 0\]
Setting \[x + 1 = 0\], which leads to \[x = -1\]

These roots, \[x = 0\] and \[x = -1\], are the critical points that determine the boundaries of the solution intervals for the inequality. They split the x-axis into distinct intervals where the inequality can be tested further to understand solution regions.

Interval Testing

Once the roots \[x = 0\] and \[x = -1\] are identified, we can use interval testing to see which parts of the number line satisfy our inequality. The roots divide the number line into three key intervals: