To determine whether the ordered pair [tex]\((6, 4)\)[/tex] is a solution to the inequality [tex]\( y < \frac{3}{4}x - 3 \)[/tex], we need to compare the [tex]\(y\)[/tex]-value of the point to the value obtained from the expression [tex]\( \frac{3}{4}x - 3 \)[/tex].
Given the inequality:
[tex]\[ y < \frac{3}{4}x - 3 \][/tex]
Let's substitute [tex]\(x = 6\)[/tex] into the expression on the right-hand side of the inequality to find its value:
[tex]\[ \frac{3}{4}(6) - 3 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 6\)[/tex]:
[tex]\[ \frac{3}{4} \times 6 = \frac{18}{4} = 4.5 \][/tex]
Next, subtract 3 from 4.5:
[tex]\[ 4.5 - 3 = 1.5 \][/tex]
Now, we know that the expression [tex]\(\frac{3}{4}x - 3\)[/tex] evaluates to 1.5 when [tex]\(x = 6\)[/tex].
So, we compare the [tex]\(y\)[/tex]-value of the ordered pair [tex]\((6, 4)\)[/tex] to this value 1.5:
- If [tex]\(y < 1.5\)[/tex], the point [tex]\((6, 4)\)[/tex] is below the line represented by the inequality.
- If [tex]\(y = 1.5\)[/tex], the point [tex]\((6, 4)\)[/tex] is on the line.
- If [tex]\(y > 1.5\)[/tex], the point [tex]\((6, 4)\)[/tex] is above the line.
Given that [tex]\(y = 4\)[/tex], we compare it with 1.5:
- [tex]\(4 > 1.5\)[/tex]
Thus, the [tex]\(y\)[/tex]-value 4 is greater than 1.5, meaning that the point [tex]\((6, 4)\)[/tex] is above the line represented by the inequality [tex]\( y < \frac{3}{4}x - 3 \)[/tex].
Therefore, the ordered pair [tex]\((6, 4)\)[/tex] is not a solution to the inequality because it is above the line.
The correct answer is:
No, because (6, 4) is above the line.
