Tell whether (-8, -2/3) is a solution of y + 1 < -1/6(x + 6). It ___ a solution.
Understand the Problem
The question is asking whether the point (-8, -2/3) satisfies the inequality y + 1 < -1/6(x + 6). To solve this, we will substitute the values of x and y into the inequality and check if the statement holds true.
Answer
It is not a solution.
Answer for screen readers
The point $(-8, -\frac{2}{3})$ is not a solution of the inequality.
Steps to Solve
- Substitute the point into the inequality
We are given the point $(-8, -\frac{2}{3})$ and we need to substitute $x = -8$ and $y = -\frac{2}{3}$ into the inequality $y + 1 < -\frac{1}{6}(x + 6)$.
So we replace $y$ and $x$:
$$ -\frac{2}{3} + 1 < -\frac{1}{6}(-8 + 6) $$
- Simplify the left side of the inequality
Calculating the left side:
$$ -\frac{2}{3} + 1 = -\frac{2}{3} + \frac{3}{3} = \frac{1}{3} $$
So the left side becomes:
$$ \frac{1}{3} < -\frac{1}{6}(-2) $$
- Simplify the right side of the inequality
Calculating the right side:
$$ -\frac{1}{6}(-2) = \frac{2}{6} = \frac{1}{3} $$
- Compare both sides of the inequality
Now we check the final inequality:
$$ \frac{1}{3} < \frac{1}{3} $$
This is not true since $\frac{1}{3}$ is equal to $\frac{1}{3}$, not less than.
The point $(-8, -\frac{2}{3})$ is not a solution of the inequality.
More Information
The solution shows that the left side is equal to the right side, indicating the point does not satisfy the strict inequality.
Tips
- Misinterpreting inequality: Some might confuse equality with the strict inequality, assuming that if both sides are equal, the inequality holds true.
- Simplifying incorrectly: Sometimes calculations with fractions can cause errors; it's important to be careful with arithmetic.
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