To solve the inequality \(-\frac{2}{3}x - 10 < \frac{1}{3}\), follow these steps:
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Isolate the term with \(x\): First, add \(10\) to both sides of the inequality:
\[ -\frac{2}{3}x < \frac{1}{3} + 10 \]
Convert \(10\) to a fraction with a common denominator of \(3\):
\[ 10 = \frac{30}{3} \]
So, we have:
\[ -\frac{2}{3}x < \frac{1}{3} + \frac{30}{3} = \frac{31}{3} \]
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Eliminate the coefficient of \(x\): Now, divide both sides by \(-\frac{2}{3}\). Remember that dividing by a negative number reverses the inequality:
\[ x > \frac{31}{3} \div -\frac{2}{3} \]
Dividing fractions can be done by multiplying by the reciprocal:
\[ \frac{31}{3} \div -\frac{2}{3} = \frac{31}{3} \times -\frac{3}{2} = -\frac{31}{2} \]
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Conclusion: Therefore, the solution to the inequality is:
\[ x > -\frac{31}{2} \]
In interval notation, the solution is:
\[ \left(-\frac{31}{2}, \infty\right) \]