Question

Solve the given inequality algebraically.

$x^4>1$

Short answer

Required solution set is$(-\infty ,-1)\cup (1,\infty )$

Step-by-step solution

Step 1. Given information

The given inequality is :$x^4>1$

We have to solve for x.

Step 2. Finding zeroes

Zeroes of the inequality $f\left(x\right)=x^4-1>0$are

$$\begin{gathered} x^4-1=0 \\ (x^2+1)(x^2-1)=0 \end{gathered}$$

$x^2$cannot be equal to $-1$

Therefore , $x^2=1$and hence

$$\begin{gathered} x=1 \\ x=-1 \end{gathered}$$

Step 3. Dividing real number line into 3 intervals

Now we use the zeroes to separate the real number line into intervals$(-\infty ,-1),(-1,1),(1,\infty )$

Step 4. Selecting a test number in each interval 

Now we select a test number in each interval found in Step 3 and evaluate at each number to determine if $f\left(x\right)=x^4-1=0$is positive or negative.

In the interval $(-\infty ,-1)$we chose -2, where f is positive

In the interval $(-1,1)$ we chose 0 , where f is negative

In the interval $(1,\infty )$we chose 2 , where f is positve

We know that our required inequality is$f\left(x\right)>0$

Here the inequality is not strict $(\ge or\le )$so we have to exclude the solutions of $f\left(x\right)=0$in the solution set.

So we want the interval where f is positive.

So required solution set is$(-\infty ,-1)\cup (1,\infty )$