Question

solve for x and graph the solution.
-(x - 1)(x + 1) < 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.
submit

Explanation:

Step1: Simplify the inequality

Multiply both sides by - 1 (and reverse the inequality sign). We get \((x - 1)(x + 1)>0\). Expand it: \(x^{2}-1>0\).

Step2: Find the roots

Set \(x^{2}-1 = 0\). Using the formula \(a^{2}-b^{2}=(a - b)(a + b)\), we have \((x - 1)(x + 1)=0\). The roots are \(x = 1\) and \(x=-1\).

Step3: Analyze the sign of the quadratic function \(y=(x - 1)(x + 1)=x^{2}-1\)

The quadratic function \(y=x^{2}-1\) is a parabola opening upwards (since the coefficient of \(x^{2}\) is \(1>0\)).
We test the intervals:

• For \(x<-1\), let \(x=-2\). Then \((-2 - 1)(-2 + 1)=(-3)\times(-1)=3>0\).
• For \(-1
• For \(x>1\), let \(x = 2\). Then \((2 - 1)(2 + 1)=1\times3 = 3>0\).

Answer:

The solution of the inequality \(-(x - 1)(x + 1)<0\) is \(x<-1\) or \(x>1\). On the number - line, we plot open circles at \(x=-1\) and \(x = 1\), and shade the regions to the left of \(x=-1\) and to the right of \(x = 1\).