Solve the Inequality for x (3-x)(x+1)(x+5)> 0
The given problem is asking to find all values of the variable x such that the product of three linear expressions, (3-x), (x+1), and (x+5), is greater than zero. This involves determining the intervals on the number line where the product of these expressions is positive, taking into account the points where each expression is equal to zero (which will be the boundaries of these intervals) and the behavior of the product around those points.
$\left(\right. 3 - x \left.\right) \left(\right. x + 1 \left.\right) \left(\right. x + 5 \left.\right) > 0$
Answer
Solution:
Step 1
Identify the roots of the inequality by setting each factor to zero:
- $3 - x = 0$
- $x + 1 = 0$
- $x + 5 = 0$
Step 2
Find the value of $x$ for the equation $3 - x = 0$.
Step 2.1
Begin by setting $3 - x$ to zero: $3 - x = 0$.
Step 2.2
Solve for $x$:
Step 2.2.1
Subtract $3$ from both sides to get $-x = -3$.
Step 2.2.2
Divide by $-1$ to isolate $x$:
Step 2.2.2.1
Divide both sides by $-1$: $\frac{-x}{-1} = \frac{-3}{-1}$.
Step 2.2.2.2
Simplify to find $x$: $x = \frac{-3}{-1}$.
Step 2.2.2.3
Calculate the result: $x = 3$.
Step 3
Solve for $x$ in the equation $x + 1 = 0$.
Step 3.1
Set $x + 1$ to zero: $x + 1 = 0$.
Step 3.2
Subtract $1$ to find $x$: $x = -1$.
Step 4
Solve for $x$ in the equation $x + 5 = 0$.
Step 4.1
Set $x + 5$ to zero: $x + 5 = 0$.
Step 4.2
Subtract $5$ to find $x$: $x = -5$.
Step 5
Determine the intervals based on the roots: $x = 3, -1, -5$.
Step 6
Create test intervals from the roots:
- $x < -5$
- $-5 < x < -1$
- $-1 < x < 3$
- $x > 3$
Step 7
Test values from each interval in the original inequality.
Step 7.1
For $x < -5$, choose $x = -8$.
Step 7.2
For $-5 < x < -1$, choose $x = -3$.
Step 7.3
For $-1 < x < 3$, choose $x = 0$.
Step 7.4
For $x > 3$, choose $x = 6$.
Step 8
Determine which intervals satisfy the inequality.
Step 9
Express the solution in different forms:
- Inequality Form: $x < -5$ or $-1 < x < 3$
- Interval Notation: $(-\infty, -5) \cup (-1, 3)$
Knowledge Notes:
To solve an inequality involving a product of factors, we must first determine the roots by setting each factor equal to zero. These roots divide the number line into intervals. We then test a value from each interval to see if it satisfies the inequality. The sign of the product changes at each root, so we only need to test one value per interval. The solution to the inequality is the union of the intervals that make the inequality true.
In this case, the inequality is a product of linear factors, which means that the roots are the values of $x$ that make each factor zero. The inequality is satisfied for intervals where the product of the factors is positive. We use the roots to define the intervals and then test points from each interval to determine where the inequality holds true.
The solution can be expressed in inequality form, showing the ranges of $x$ that satisfy the inequality, or in interval notation, which provides a concise way to represent the solution set. Interval notation uses parentheses to denote open intervals and brackets for closed intervals. Since inequalities do not include the endpoints (the roots in this case), we use parentheses.
