Question

equations and inequalities
solving a word problem using a two - step linear inequality and describing...
uel has a tree in his backyard. the tree grows 25 centimeters each year. since the tree started growing, miguel has trimmed 16 centimeters off its height.
the goal for the tree’s height is given by (25x - 16>234), 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.
(the number line and graphing tools are shown here, omitted as per ocr text processing rules)
(b) choose and complete the statement that best describes the solution.
○ to meet the goal, the tree must be more than (square) centimeters tall. this will take less than (square) years.
○ to meet the goal, the tree must be more than (square) centimeters tall. this will take more than (square) years.

Explanation:

Part (a)

Step 1: Add 16 to both sides

To isolate the term with \( x \), we add 16 to both sides of the inequality \( 25x - 16 > 234 \).
\[
25x - 16 + 16 > 234 + 16
\]
\[
25x > 250
\]

Step 2: Divide both sides by 25

Now, we divide both sides by 25 to solve for \( x \).
\[
\frac{25x}{25} > \frac{250}{25}
\]
\[
x > 10
\]
To graph this on the number line, we use an open circle at 10 (since \( x \) is greater than 10, not greater than or equal to) and draw an arrow to the right.

Part (b)

The inequality \( 25x - 16 > 234 \) models the tree's height. The right - hand side of the original inequality, 234, is the goal height (in centimeters) for the tree. From the solution \( x > 10 \), \( x \) is the number of years the tree grows. So, to meet the goal, the tree must be more than 234 centimeters tall, and this will take more than 10 years.

Final Answers:

Part (a)

The solution to the inequality is \( x > 10 \). For the number - line graph, place an open circle at 10 and draw an arrow to the right.

Part (b)

The correct statement is: "To meet the goal, the tree must be more than \(\boldsymbol{234}\) centimeters tall. This will take more than \(\boldsymbol{10}\) years."

Answer:

The inequality \( 25x - 16 > 234 \) models the tree's height. The right - hand side of the original inequality, 234, is the goal height (in centimeters) for the tree. From the solution \( x > 10 \), \( x \) is the number of years the tree grows. So, to meet the goal, the tree must be more than 234 centimeters tall, and this will take more than 10 years.

Final Answers:

Part (a)

The solution to the inequality is \( x > 10 \). For the number - line graph, place an open circle at 10 and draw an arrow to the right.

Part (b)

The correct statement is: "To meet the goal, the tree must be more than \(\boldsymbol{234}\) centimeters tall. This will take more than \(\boldsymbol{10}\) years."