Solve the non-linear inequality:
x^3-5x^2-14x<0
To solve the non-linear inequality \(x^3 - 5x^2 - 14x < 0\), we need to find the interval(s) of \(x\) that satisfy the inequality.
Here's how to solve it step by step:
Step 1: Factorize the expression on the left side to make it easier to work with.
\(x^3 - 5x^2 - 14x < 0\)
\(x(x^2 - 5x - 14) < 0\)
Step 2: Solve for \(x\) to find the critical points.
Set each factor equal to zero:
\(x = 0\)
\(x^2 - 5x - 14 = 0\)
Solve the quadratic equation by factoring, using the quadratic formula, or completing the square. The solutions are:
\(x = -2\) and \(x = 7\)
So, the critical points are \(x = 0\), \(x = -2\), and \(x = 7\).
Step 3: Create a number line to determine the intervals.
Place the critical points on a number line in ascending order:
\(-2, 0, 7\)
Step 4: Test each interval.
To determine the sign of the expression \(x(x^2 - 5x - 14)\) in each interval, pick a test value from each interval and evaluate the expression:
For \(x < -2\), choose \(x = -3\)
\((-3)((-3)^2 - 5(-3) - 14) = 51 > 0\)
For \(-2 < x < 0\), choose \(x = -1\)
\((-1)((-1)^2 - 5(-1) - 14) = 18 < 0\)
For \(0 < x < 7\), choose \(x = 5\)
\(5(5^2 - 5(5) - 14) = -240 < 0\)
For \(x > 7\), choose \(x = 8\)
\(8(8^2 - 5(8) - 14) = 480 > 0\)
Step 5: Determine the solution.
On the number line, the intervals where the expression is less than zero (\(< 0\)) are:
\(-2 < x < 0\) and \(0 < x < 7\)
Therefore, the solution to the inequality \(x^3 - 5x^2 - 14x < 0\) is:
\(-2 < x < 0\) or \(0 < x < 7\)