Question

A (a^2-3)+5a>a^3+2 (a-2)

Answer 1

a(a²-3)+5a>a³+2(a-2) =>

a³-3a+5a>a³+2a-4 =>

a³+2a-a³-2a>-4 =>

0>-4

Ответ: бесконечное множество решений

Answer 2

To solve the inequality $A: (a^2-3)+5a>a^3+2(a-2)$, we'll first simplify the expression and then find the values of $a$ that satisfy the inequality.

Step 1: Simplify the expression $A: (a^2-3)+5a > a^3 + 2(a-2)$

Expand the expressions on both sides:

$A: a^2 - 3 + 5a > a^3 + 2a - 4$

Step 2: Move all terms to one side of the inequality to set it to zero:

$A: a^3 - a^2 - 5a + 2a - 3 + 4 > 0$

$A: a^3 - a^2 - 3a + 1 > 0$

Step 3: Find the critical points where the expression equals zero. These are the values of $a$ that make the inequality invalid. To find the critical points, set the expression inside the inequality to zero:

$a^3 - a^2 - 3a + 1 = 0$

Step 4: Solve for $a$. Unfortunately, finding exact solutions for this cubic equation can be complex, and it might not have rational solutions. Since the inequality involves real numbers, we can use numerical methods or graphing software to find approximate solutions.

Approximate solutions are: $a \approx -0.695$, $a \approx 0.347$, $a \approx 2.348$

Step 5: Plot the inequality on a number line and check the intervals:

• Interval 1: $a < -0.695$
• Interval 2: $-0.695 < a < 0.347$
• Interval 3: $0.347 < a < 2.348$
• Interval 4: $a > 2.348$

Step 6: Test the intervals to find the solution set:

• Test a value in Interval 1, e.g., $a = -1$: $a^3 - a^2 - 3a + 1 = -1 - 1 + 3 + 1 = 2 > 0$. True.
• Test a value in Interval 2, e.g., $a = 0$: $a^3 - a^2 - 3a + 1 = 1 > 0$. True.
• Test a value in Interval 3, e.g., $a = 1$: $a^3 - a^2 - 3a + 1 = -2 < 0$. False.
• Test a value in Interval 4, e.g., $a = 3$: $a^3 - a^2 - 3a + 1 = 13 > 0$. True.

Step 7: Write down the solution set:

$A: -0.695 < a < 0$ or $a > 2.348$

So, the solution to the inequality is $-0.695 < a < 0$ or $a > 2.348$.