Solve algebraically. \[ 7(t-2)+10 t<2(5+5 t) \] Write the solution in set-builder notation. \[ \{t \square \] (Type an inequality. Use integers or fractions for any numbers in the inequality.) The solution in interval notation is $\square$. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
Solution
Step 1 ：Distribute the 7 and 2 on the left and right side of the inequality respectively: \(7t - 14 + 10t < 10 + 10t\).
Step 2 ：Combine like terms: \(17t - 14 < 10 + 10t\).
Step 3 ：Subtract 10t from both sides to isolate t: \(7t - 14 < 10\).
Step 4 ：Add 14 to both sides to isolate t: \(7t < 24\).
Step 5 ：Divide both sides by 7 to solve for t: \(t < \frac{24}{7}\).
Step 6 ：The solution to the inequality is \(t < \frac{24}{7}\). This means that the set of all t that satisfy the inequality is \(\{t | t < \frac{24}{7}\}\).
Step 7 ：In interval notation, this is represented as \((-∞, \frac{24}{7})\).
Step 8 ：\(\boxed{\text{In set-builder notation, the solution is } \{t | t < \frac{24}{7}\}. \text{ In interval notation, the solution is } (-∞, \frac{24}{7})}\)
