Question

diane is pumping water into a tank at a rate of 17 liters per minute. since she started pumping water, 16 liters has splashed out of the tank. her goal is given by 17x - 16 < 52, where x is the number of minutes she pumps water into the tank. complete the parts below. (a) solve the given inequality and graph the solution on the number line below. (b) choose and complete the statement that best describes the solution. to meet the goal, the tank will have less than liters of water in it. this will take more than minutes. to meet the goal, the tank will have less than liters of water in it.

Explanation:

Step1: Solve the inequality \(17x - 16 < 52\)

Add 16 to both sides of the inequality: \(17x - 16 + 16 < 52 + 16\), which simplifies to \(17x < 68\).

Step2: Divide both sides by 17

Divide each side by 17: \(\frac{17x}{17} < \frac{68}{17}\), so \(x < 4\).

For graphing on the number line (not fully detailed here as the main solve is \(x < 4\)): We would use an open circle at 4 (since \(x\) is less than, not less than or equal to) and shade to the left.

For part (b), the goal inequality is \(17x - 16 < 52\), which represents the amount of water in the tank (since \(17x\) is the water pumped in, minus 16 splashed out, and it's less than 52). So the tank will have less than 52 liters. And from the solution \(x < 4\), it will take less than 4 minutes (wait, the original option might have a typo, but based on the inequality solution \(x < 4\), so "To meet the goal, the tank will have less than 52 liters of water in it. This will take less than 4 minutes." But following the problem's options, likely the first blank is 52, the second is 4 (but need to check the options again, but from the inequality solve, \(x < 4\) and the right-hand side of the inequality is 52, so the tank has less than 52 liters, and time is less than 4 minutes).

Answer:

(a) The solution to the inequality is \(x < 4\). For graphing, an open circle at 4 and shade left.
(b) To meet the goal, the tank will have less than \(\boldsymbol{52}\) liters of water in it. This will take less than \(\boldsymbol{4}\) minutes.