Let's solve this problem step-by-step to determine if the point [tex]\((2,1)\)[/tex] satisfies the inequality [tex]\( y < -x + 3 \)[/tex].
1. Substitute the Coordinates:
We have the inequality [tex]\( y < -x + 3 \)[/tex] and the given point [tex]\((2,1)\)[/tex]. We need to substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality to see if it holds true.
2. Perform the Substitution:
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality:
[tex]\[
1 < -2 + 3
\][/tex]
3. Simplify the Right-Hand Side:
Simplify the expression on the right-hand side:
[tex]\[
-2 + 3 = 1
\][/tex]
4. Compare Both Sides:
Now we have:
[tex]\[
1 < 1
\][/tex]
5. Evaluate the Inequality:
Clearly, [tex]\( 1 \)[/tex] is not less than [tex]\( 1 \)[/tex]. Thus, the inequality does not hold true for the point [tex]\((2,1)\)[/tex].
Therefore, the point [tex]\((2,1)\)[/tex] does not satisfy the inequality [tex]\( y < -x + 3 \)[/tex]. Since the inequality is not true for the given point, [tex]\((2,1)\)[/tex] is not in the solution set of the inequality [tex]\( y < -x + 3 \)[/tex].
