Q. 12 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.    

$x^2-1<0$

Step-by-Step Solution

Answer

The solution set is $(-1,1)$.

Step 1. Given Information

The given inequality is $x^2-1<0$.

Step 2. Find the intercepts and vertex.

• The value of $f\left(0\right)=-1$.
• So, the y-intercept is $(0,-1)$.
• Factor the equation.

$(x+1)(x-1)=0$

• Equate the factors to 0.

$\begin{array}{rcl}x+1& =& 0\\ x& =& -1\\ x-1& =& 0\\ x& =& 1\end{array}$

• So, the x-intercepts are $(1,0),(-1,0)$.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-0}{2\left(1\right)}=0$
• So, the vertex is $(0,-1)$.

Step 3. Plot the graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.  

Step 4. Find the solution set

• From the graph, it is observed that for x belonging to $(-1,1)$, $y<0$.
• So, the solution set of the given inequality is $(-1,1)$.