Solve the Inequality for x (x+3)/((x-4)^2(x^2-4))< =0
The given mathematical problem is asking to find the range of values for the variable x that makes the given inequality true. Specifically, the inequality is (x+3)/((x-4)^2*(x^2-4))<=0, which is a rational inequality because it involves a fraction with polynomials in the numerator and denominator. The challenge is to determine the intervals on the x-axis where the expression is less than or equal to zero, while considering the points where the expression is undefined due to division by zero.
$\frac{\left(\right. x + 3 \left.\right)}{\left(\left(\right. x - 4 \left.\right)\right)^{2} \left(\right. x^{2} - 4 \left.\right)} \leq 0$
Answer
Solution:
Step:1
Identify critical values where the function changes sign by equating each factor to $0$ and solving for $x$:
$x + 3 = 0$ $(x - 4)^2 = 0$ $x^2 - 4 = 0$
Step:2
Isolate $x$ by subtracting $3$:
$x = -3$
Step:3
Solve $x - 4 = 0$:
$x = 4$
Step:4
Solve $x^2 = 4$:
$x = \pm 2$
Step:5
Simplify the square root:
$x = \pm \sqrt{2^2}$
Step:6
Extract the square root:
$x = \pm 2$
Step:7
List all solutions:
$x = -3, 4, 2, -2$
Step:8
Determine the domain of $\frac{x + 3}{(x - 4)^2 (x^2 - 4)}$ by setting the denominator to $0$ and solving for $x$.
Step:9
Solve for $x$ in $(x - 4)^2 = 0$ and $x^2 - 4 = 0$.
Step:10
Find the domain:
$x \neq 4, 2, -2$
Step:11
Create test intervals based on the critical values:
$x < -3$, $-3 < x < -2$, $-2 < x < 2$, $2 < x < 4$, $x > 4$
Step:12
Test values from each interval in the original inequality to determine which intervals satisfy the inequality.
Step:13
Conclude which intervals make the inequality true:
$x \leq -3$ or $-2 < x < 2$
Step:14
Present the solution in different forms:
Inequality Form: $x \leq -3$ or $-2 < x < 2$ Interval Notation: $(-\infty, -3] \cup (-2, 2)$
Knowledge Notes:
To solve an inequality involving rational expressions, one must consider the following steps:
Factorization: Break down the expression into its simplest factors to identify critical values where the function may change sign.
Critical Values: Solve for the values of $x$ that make each factor equal to zero. These values are where the function is undefined or where it may change sign.
Test Intervals: Between each pair of critical values, the sign of the function is consistent. Choose test values from each interval to determine the sign of the function in that interval.
Sign Analysis: Use the test values to determine whether the function is positive or negative in each interval.
Solution Set: The solution to the inequality consists of all intervals where the function satisfies the inequality (is non-positive in this case).
Domain: The domain of the function excludes values that make the denominator zero since division by zero is undefined.
Interval Notation: The solution can be expressed in interval notation, which is a concise way to represent ranges of values for which the inequality holds true.
Inequalities involving rational expressions can have solutions that are single values, ranges of values, or a combination of both. It is important to include the critical values in the solution set if the original inequality includes equality (i.e., $\leq$ or $\geq$). When presenting the final answer, it is common to use both inequality notation and interval notation for clarity.
