Certainly! Let's solve the inequality step-by-step:
The given inequality is:
[tex]\[ x^2 - 2x > x^2 + 3x - 5 \][/tex]
1. Simplify the Inequality:
First, we want to move all the terms to one side of the inequality to simplify it. Let's subtract [tex]\( x^2 + 3x - 5 \)[/tex] from both sides of the inequality:
[tex]\[
(x^2 - 2x) - (x^2 + 3x - 5) > 0
\][/tex]
2. Combine Like Terms:
Now, combine like terms on the left-hand side:
[tex]\[
x^2 - 2x - x^2 - 3x + 5 > 0
\][/tex]
Simplifying this, we get:
[tex]\[
-5x + 5 > 0
\][/tex]
3. Isolate the Variable [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], we can first add [tex]\(5x\)[/tex] to both sides:
[tex]\[
5 > 5x
\][/tex]
Then, divide by 5:
[tex]\[
1 > x
\][/tex]
or equivalently:
[tex]\[
x < 1
\][/tex]
4. Conclusion:
The solution to the inequality [tex]\( x^2 - 2x > x^2 + 3x - 5 \)[/tex] is:
[tex]\[
x < 1
\][/tex]
Therefore, the inequality holds true for all [tex]\( x \)[/tex] in the interval:
[tex]\[
(-\infty, 1)
\][/tex]
This is our final solution.
