Question

for each value of v, determine whether it is a solution to 69 < -8v + 5.
is it a solution?
yes \t no
v \t \t
-9 \t circ \t circ
10 \t circ \t circ
-2 \t circ \t circ
-8 \t circ \t circ

Explanation:

To determine if a value of \( v \) is a solution to the inequality \( 69 < -8v + 5 \), we first solve the inequality for \( v \):

Step 1: Subtract 5 from both sides

Subtract 5 from both sides of the inequality to isolate the term with \( v \):
\[
69 - 5 < -8v + 5 - 5
\]
\[
64 < -8v
\]

Step 2: Divide both sides by -8 (and reverse the inequality sign)

Divide both sides by -8. Remember that when dividing or multiplying an inequality by a negative number, the inequality sign reverses:
\[
\frac{64}{-8} > \frac{-8v}{-8}
\]
\[
-8 > v
\]
or equivalently,
\[
v < -8
\]

Now we check each value of \( v \) to see if it satisfies \( v < -8 \):

1. For \( v = -9 \):

Check if \( -9 < -8 \). Since \( -9 \) is less than \( -8 \), this is true. So \( -9 \) is a solution (Yes).

1. For \( v = 10 \):

Check if \( 10 < -8 \). Since \( 10 \) is greater than \( -8 \), this is false. So \( 10 \) is not a solution (No).

1. For \( v = -2 \):

Check if \( -2 < -8 \). Since \( -2 \) is greater than \( -8 \), this is false. So \( -2 \) is not a solution (No).

1. For \( v = -8 \):

Check if \( -8 < -8 \). Since \( -8 \) is not less than \( -8 \) (it is equal), this is false. So \( -8 \) is not a solution (No).

Final Answers (Filling the Table):

\( v \)	Is it a solution? Yes	Is it a solution? No
\( 10 \)		\( \circ \)
\( -2 \)		\( \circ \)
\( -8 \)		\( \circ \)

(Mark "Yes" for \( -9 \) and "No" for \( 10 \), \( -2 \), \( -8 \) as shown above.)

Answer:

To determine if a value of \( v \) is a solution to the inequality \( 69 < -8v + 5 \), we first solve the inequality for \( v \):

Step 1: Subtract 5 from both sides

Subtract 5 from both sides of the inequality to isolate the term with \( v \):
\[
69 - 5 < -8v + 5 - 5
\]
\[
64 < -8v
\]

Step 2: Divide both sides by -8 (and reverse the inequality sign)

Divide both sides by -8. Remember that when dividing or multiplying an inequality by a negative number, the inequality sign reverses:
\[
\frac{64}{-8} > \frac{-8v}{-8}
\]
\[
-8 > v
\]
or equivalently,
\[
v < -8
\]

Now we check each value of \( v \) to see if it satisfies \( v < -8 \):

1. For \( v = -9 \):

Check if \( -9 < -8 \). Since \( -9 \) is less than \( -8 \), this is true. So \( -9 \) is a solution (Yes).

1. For \( v = 10 \):

Check if \( 10 < -8 \). Since \( 10 \) is greater than \( -8 \), this is false. So \( 10 \) is not a solution (No).

1. For \( v = -2 \):

Check if \( -2 < -8 \). Since \( -2 \) is greater than \( -8 \), this is false. So \( -2 \) is not a solution (No).

1. For \( v = -8 \):

Check if \( -8 < -8 \). Since \( -8 \) is not less than \( -8 \) (it is equal), this is false. So \( -8 \) is not a solution (No).

Final Answers (Filling the Table):

\( v \)	Is it a solution? Yes	Is it a solution? No
\( 10 \)		\( \circ \)
\( -2 \)		\( \circ \)
\( -8 \)		\( \circ \)

(Mark "Yes" for \( -9 \) and "No" for \( 10 \), \( -2 \), \( -8 \) as shown above.)