Question

practice solving problems involving two - step inequalities. which number line shows the solution to (11x + 14lt - 87)?
a
(leftarrow\begin{array}{ccccccccc}-6&-4&-2&0&2&4&6&8&10end{array}
ightarrow) (with an open circle at 2 and a line to the right)

b
(leftarrow\begin{array}{ccccccccc}-6&-4&-2&0&2&4&6&8&10end{array}
ightarrow) (with an open circle at - 2 and a line to the right)

c
(leftarrow\begin{array}{ccccccccc}-6&-4&-2&0&2&4&6&8&10end{array}
ightarrow) (with an open circle at 0 and a line to the left)

d
(leftarrow\begin{array}{ccccccccc}-6&-4&-2&0&2&4&6&8&10end{array}
ightarrow) (with an open circle at - 2 and a line to the left)

Explanation:

Step1: Solve the inequality \(11x + 14 < -87\)

Subtract 14 from both sides: \(11x + 14 - 14 < -87 - 14\)
Simplify: \(11x < -101\)

Step2: Divide both sides by 11

\(\frac{11x}{11} < \frac{-101}{11}\)
Simplify: \(x < -\frac{101}{11} \approx -9.18\)? Wait, no, wait, I made a mistake. Wait, -87 -14 is -101? Wait, no, -87 -14: 87 +14 is 101, so -87 -14 = -101. Then 11x < -101, so x < -101/11 ≈ -9.18? But the number lines have -6, -4, -2, 0, 2... Wait, maybe I miscalculated. Wait, let's check again. Wait, the inequality is \(11x + 14 < -87\). Subtract 14: \(11x < -87 -14 = -101\). Then x < -101/11 ≈ -9.18. But the number lines start at -6? Wait, maybe the problem was written wrong? Wait, no, maybe I misread the inequality. Wait, maybe it's \(11x + 14 < -8\)? No, the user wrote \(11x + 14 < -87\). Wait, but the number lines have -6, -4, -2, 0, 2... So maybe there's a typo, but assuming the problem is correct, or maybe I made a mistake. Wait, wait, 11x +14 < -87. Let's do it again:

11x +14 < -87

Subtract 14: 11x < -87 -14 = -101

Divide by 11: x < -101/11 ≈ -9.18. But the number lines have -6 as the leftmost, so maybe the inequality is \(11x + 14 < -8\)? Let's try that. 11x +14 < -8: 11x < -22: x < -2. Ah! That makes sense. So maybe the original problem was \(11x +14 < -8\)? Because then x < -2. Let's check: 11x +14 < -8: 11x < -22: x < -2. Then the number line with an open circle at -2 and pointing to the left (since x < -2). Looking at the options: D has an open circle at -2 and pointing left. Wait, maybe the user mistyped -87 as -8? Because -87 gives a solution left of -9, which isn't on the number lines. So assuming it's a typo and the inequality is \(11x +14 < -8\), then:

Step1: Solve \(11x +14 < -8\)

Subtract 14: \(11x < -8 -14 = -22\)

Step2: Divide by 11: \(x < -2\)

So the solution is x < -2, which is a number line with an open circle at -2 and the arrow pointing to the left (since x is less than -2). Looking at the options:

A: Open circle at 2, right arrow.

B: Open circle at -2, right arrow.

C: Open circle at 0, left arrow.

D: Open circle at -2, left arrow.

So if x < -2, the arrow points left from -2 (open circle). So option D.

Assuming the inequality was supposed to be \(11x +14 < -8\) (since -87 gives a solution not on the number lines), the correct number line is D.

Answer:

D (with open circle at -2 and arrow pointing left)