Questions: Solve the inequality. (Enter your answer using interval notation) (x+5)^2(x+3)(x-1)>0 (-∞,-3) ∪ (1, ∞)

Solution Steps

To solve the inequality \((x+5)^{2}(x+3)(x-1)>0\), we need to determine the intervals where the expression is positive. First, identify the critical points by setting each factor equal to zero: \(x+5=0\), \(x+3=0\), and \(x-1=0\). These give the critical points \(x=-5\), \(x=-3\), and \(x=1\). Next, test the sign of the expression in the intervals determined by these critical points: \((-∞, -5)\), \((-5, -3)\), \((-3, 1)\), and \((1, ∞)\). Since \((x+5)^2\) is always non-negative, focus on the sign changes caused by \((x+3)\) and \((x-1)\).

To solve the inequality \((x+5)^{2}(x+3)(x-1)>0\), we first identify the critical points by setting each factor equal to zero:

1. \((x+5)^2 = 0\) gives \(x = -5\).
2. \((x+3) = 0\) gives \(x = -3\).
3. \((x-1) = 0\) gives \(x = 1\).

Thus, the critical points are \(x = -5\), \(x = -3\), and \(x = 1\).

The critical points divide the real number line into the following intervals:

1. \((- \infty, -5)\)
2. \((-5, -3)\)
3. \((-3, 1)\)
4. \((1, \infty)\)

We will test a point from each interval to determine where the inequality \((x+5)^{2}(x+3)(x-1)>0\) holds true.

1. Interval \((- \infty, -5)\): Choose \(x = -6\). \[ ((-6+5)^2)(-6+3)(-6-1) = (1)(-3)(-7) = -21 < 0 \] The inequality does not hold.

2. Interval \((-5, -3)\): Choose \(x = -4\). \[ ((-4+5)^2)(-4+3)(-4-1) = (1)(-1)(-5) = 5 > 0 \] The inequality holds.

3. Interval \((-3, 1)\): Choose \(x = 0\). \[ ((0+5)^2)(0+3)(0-1) = (25)(3)(-1) = -75 < 0 \] The inequality does not hold.

4. Interval \((1, \infty)\): Choose \(x = 2\). \[ ((2+5)^2)(2+3)(2-1) = (49)(5)(1) = 245 > 0 \] The inequality holds.

The factor \((x+5)^2\) has an even multiplicity, meaning the sign of the expression does not change at \(x = -5\). Therefore, the interval \((-5, -3)\) is valid, but \(-5\) itself is not included.

Final Answer