Certainly! Let's solve the inequality [tex]\( x^2 - 10x + 1 < 20 + 5x \)[/tex] for [tex]\( x = -2 \)[/tex].
1. Substitute [tex]\( x = -2 \)[/tex] into both sides of the inequality:
- The left side of the inequality is [tex]\( x^2 - 10x + 1 \)[/tex].
- The right side of the inequality is [tex]\( 20 + 5x \)[/tex].
2. Evaluate the left side:
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( x^2 - 10x + 1 \)[/tex]:
[tex]\[
(-2)^2 - 10(-2) + 1
\][/tex]
- First, calculate [tex]\( (-2)^2 \)[/tex]:
[tex]\[
4
\][/tex]
- Next, calculate [tex]\( -10 \times (-2) \)[/tex]:
[tex]\[
20
\][/tex]
- Now, add these results together along with 1:
[tex]\[
4 + 20 + 1 = 25
\][/tex]
3. Evaluate the right side:
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( 20 + 5x \)[/tex]:
[tex]\[
20 + 5(-2)
\][/tex]
- First, calculate [tex]\( 5 \times (-2) \)[/tex]:
[tex]\[
-10
\][/tex]
- Now, add this result to 20:
[tex]\[
20 - 10 = 10
\][/tex]
4. Compare the two sides:
- The left side evaluates to 25.
- The right side evaluates to 10.
5. Determine if the inequality holds:
- We want to see if [tex]\( 25 < 10 \)[/tex].
6. Conclusion:
- Since [tex]\( 25 \)[/tex] is not less than [tex]\( 10 \)[/tex], the inequality does not hold for [tex]\( x = -2 \)[/tex].
Therefore, for [tex]\( x = -2 \)[/tex], the inequality [tex]\( x^2 - 10x + 1 < 20 + 5x \)[/tex] does not hold true.
