To solve the compound inequality \(1 < y + \frac{10}{3} \leq 5\), we'll break it down into two parts and solve them separately.
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Solve the lower part of the inequality: \[ 1 < y + \frac{10}{3} \] Subtract \(\frac{10}{3}\) from both sides: \[ 1 - \frac{10}{3} < y \] To combine the terms on the left, convert \(1\) to a fraction with a denominator of \(3\): \[ 1 = \frac{3}{3} \] Thus, we have: \[ \frac{3}{3} - \frac{10}{3} < y \] Simplifying this gives: \[ \frac{3 - 10}{3} < y \implies \frac{-7}{3} < y \]
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Solve the upper part of the inequality: \[ y + \frac{10}{3} \leq 5 \] Subtract \(\frac{10}{3}\) from both sides: \[ y \leq 5 - \frac{10}{3} \] Convert \(5\) to a fraction with a denominator of \(3\): \[ 5 = \frac{15}{3} \] Therefore, we have: \[ y \leq \frac{15}{3} - \frac{10}{3} \] Simplifying this gives: \[ y \leq \frac{15 - 10}{3} \implies y \leq \frac{5}{3} \]
Now, we combine both parts of the inequality: \[ \frac{-7}{3} < y \leq \frac{5}{3} \]
To express the solution with integers, we can convert the fractional bounds to decimal equivalents: \[ -2.33 < y \leq 1.67 \]
The integers that fall within this range are \( -2, -1, 0, 1\).
Thus, the solution in integer form is: \[ y \in {-2, -1, 0, 1} \]
Or as a compound inequality with integers: \[ -2 \leq y \leq 1 \]