Question

solve for x and graph the solution.

x² + 2x - 8 < 0

plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.

Explanation:

Step1: Factor the quadratic expression

We factor \(x^{2}+2x - 8\). Using the formula \(x^{2}+(a + b)x+ab=(x + a)(x + b)\), where \(a = 4\) and \(b=-2\) since \(4\times(-2)=-8\) and \(4+( - 2)=2\). So \(x^{2}+2x - 8=(x + 4)(x - 2)\)

Step2: Find the roots

Set \((x + 4)(x - 2)=0\). By the zero - product property \(x+4 = 0\) gives \(x=-4\) and \(x - 2=0\) gives \(x = 2\)

Step3: Test intervals

We have three intervals: \((-\infty,-4)\), \((-4,2)\) and \((2,\infty)\)

• For \(x=-5\) (in \((-\infty,-4)\)): \((-5 + 4)(-5 - 2)=(-1)\times(-7)=7>0\)
• For \(x = 0\) (in \((-4,2)\)): \((0 + 4)(0 - 2)=(4)\times(-2)=-8<0\)
• For \(x=3\) (in \((2,\infty)\)): \((3 + 4)(3 - 2)=7\times1 = 7>0\)

Answer:

The solution of the inequality \(x^{2}+2x - 8<0\) is \(-4