Answer
(-$\infty$,-1) U (1,$\infty$)
Work Step by Step
1.) Set what the inequality equals to what it originally equals, and then set it to the opposite of that number with the opposite inequality sign
$x^{2}$ + 2x$\gt$2 and $x^{2}$ + 2x$\lt$-2
2.) Solve the first inequality for x by subtracting 2 from both sides and factoring.
$x^{2}$ + 2x$\gt$2
-2 -2
$x^{2}$ + 2x - 2$\gt$0
$(x-1)^{2}$$\gt$0
3.) Solve the first inequality for x by taking the square root of both sides and adding 1
$(x-1)^{2}$$\gt$0
x-1$\gt$0
+1 +1
x$\gt$1
4.) Identify the interval of x values the inequality is identifying
x$\gt$1
5.) Solve the second inequality for x by adding two to both sides and factoring.
$x^{2}$ + 2x$\lt$-2
+2 +2
$x^{2}$ + 2x+2$\lt$0
$(x+1)^{2}$$\gt$0
6.) Solve the first inequality for x by taking the square root of both sides and subtracting 1
$(x+1)^{2}$$\gt$0
x+1$\gt$0
-1 -1
x$\gt$-1
7.) Identify the interval of x values the inequality is identifying
x$\gt$-1
8.)Combine the two intervals using a union.
(-$\infty$,-1) U (1,$\infty$)
