Question

equations and inequalities
solving a word problem using a two - step linear inequality and describing...
josh has a tree in his backyard. the tree grows 25 centimeters each year. since the tree started growing, josh has trimmed 16 centimeters off its height. the goal for the tree’s height is given by $25x - 16\geq234$, where $x$ is the number of years the tree grows.
complete the parts below.
(a) solve the given inequality and graph the solution on the number line below.
(there is a number line from 0 to 30 with marks at 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 and a set of graphing tools including a eraser, an open circle, a closed circle, an arrow, an undo button, and a clear button)
(b) choose and complete the statement that best describes the solution.
\\(\circ\\) to meet the goal, the tree must be at least \\(\square\\) centimeters tall. this will take at most \\(\square\\) years.
\\(\circ\\) to meet the goal, the tree must be at least \\(\square\\) centimeters tall. this will take at least \\(\square\\) years.
\\(\circ\\) to meet the goal, the tree must be at most \\(\square\\) centimeters tall. this will take at least \\(\square\\) years.

Explanation:

Step1: Add 16 to both sides

\(25x - 16 + 16 \geq 234 + 16\)
\(25x \geq 250\)

Step2: Divide by 25

\(\frac{25x}{25} \geq \frac{250}{25}\)
\(x \geq 10\)

Step3: Analyze inequality for part (b)

The inequality \(25x - 16 \geq 234\) means the tree’s height (\(25x -16\)) is at least 234 cm. The solution \(x \geq10\) means it takes at least 10 years. So the correct statement is: "To meet the goal, the tree must be at least 234 centimeters tall. This will take at least 10 years."

Answer:

(a) \(x \geq 10\)
(b) To meet the goal, the tree must be at least 234 centimeters tall. This will take at least 10 years.