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A student claims that the point [tex]$(15, 5)$[/tex] is a solution to the inequality [tex]$3x - 4y \geq 24$[/tex]. Is the student correct? Show or explain your reasoning.

Sagot :

To determine if the point [tex]\((15, 5)\)[/tex] satisfies the inequality [tex]\(3x - 4y \geq 24\)[/tex], we need to substitute [tex]\(x = 15\)[/tex] and [tex]\(y = 5\)[/tex] into the inequality and check if the resulting expression holds true.

1. Substitute:
[tex]\[ 3x - 4y \geq 24 \][/tex]
We substitute [tex]\(x = 15\)[/tex] and [tex]\(y = 5\)[/tex]:
[tex]\[ 3(15) - 4(5) \geq 24 \][/tex]

2. Calculate:
We calculate the left-hand side of the inequality:
[tex]\[ 3(15) = 45 \][/tex]
[tex]\[ 4(5) = 20 \][/tex]
Thus, the expression becomes:
[tex]\[ 45 - 20 \geq 24 \][/tex]

3. Simplify:
Simplify the left-hand side:
[tex]\[ 25 \geq 24 \][/tex]

4. Interpret the result:
Since [tex]\(25 \geq 24\)[/tex] is a true statement, the given point [tex]\((15, 5)\)[/tex] satisfies the inequality.

Therefore, the student is correct. The point [tex]\((15, 5)\)[/tex] is indeed a solution to the inequality [tex]\(3x - 4y \geq 24\)[/tex].
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