Question

select the correct answer. which inequality is represented by this number line? (number line with red markings at -4 and 5, and arrows extending left from -4 and right from 5) options: |2x + 1| ≥ 7; 2|x + 1| ≤ 8; |2x - 1| ≥ 9; 2|x - 1| ≤ 10

Explanation:

Step1: Identify number line inequality

The number line has closed dots at $x=-4$ and $x=6$, with arrows pointing outward. This represents $x \leq -4$ or $x \geq 6$, which is equivalent to $|x - a| \geq b$ where $a$ is the midpoint, $b$ is the distance from midpoint to endpoints.
Midpoint: $\frac{-4 + 6}{2} = 1$, distance: $6 - 1 = 5$. So the inequality is $|x - 1| \geq 5$.

Step2: Test option 1: $|2x+1| \geq 7$

Split into two inequalities:
$2x+1 \geq 7$ or $2x+1 \leq -7$
$2x \geq 6$ or $2x \leq -8$
$x \geq 3$ or $x \leq -4$ → Does not match.

Step3: Test option 2: $2|x+1| \leq 8$

Simplify: $|x+1| \leq 4$
$-4 \leq x+1 \leq 4$
$-5 \leq x \leq 3$ → Does not match.

Step4: Test option 3: $|2x-1| \geq 9$

Split into two inequalities:
$2x-1 \geq 9$ or $2x-1 \leq -9$
$2x \geq 10$ or $2x \leq -8$
$x \geq 5$ or $x \leq -4$ → Does not match.

Step5: Test option 4: $2|x-1| \leq 10$

*Correction: Rewrite our identified inequality $|x-1| \geq 5$ as $2|x-1| \geq 10$? No, recheck the number line: outward arrows mean **greater than or equal**, so the correct derived inequality is $|x-1| \geq 5$, multiply by 2: $2|x-1| \geq 10$? Wait, no—re-solve option 1 correctly:
Wait re-solve $|2x+1| \geq7$:
$2x+1 \geq7$ → $2x\geq6$ → $x\geq3$; $2x+1\leq-7$ → $2x\leq-8$ → $x\leq-4$. No.
Re-solve $|2x-1|\geq9$:
$2x-1\geq9$ → $2x\geq10$ → $x\geq5$; $2x-1\leq-9$ → $2x\leq-8$ → $x\leq-4$. No.
Wait the number line is $x\leq-4$ or $x\geq6$. Let's plug $x=6$ into each option:
Option1: $|2*6+1|=13\geq7$ (true), but $x=6$ is part of the solution, but the upper bound is $x\geq3$, not $x\geq6$.
Option3: $|2*6-1|=11\geq9$ (true), upper bound $x\geq5$, not $x\geq6$.
Wait let's build the inequality from $x\leq-4$ or $x\geq6$:
First, shift to center at $\frac{-4+6}{2}=1$. Let $y=x-1$, then $y\leq-5$ or $y\geq5$, so $|y|\geq5$, so $|x-1|\geq5$. Multiply both sides by 2: $2|x-1|\geq10$. But the option is $2|x-1|\leq10$ which is the opposite. Wait, did I misread the number line? The arrows are **outward**, so it's all numbers less than or equal to -4, or greater than or equal to 6. Now solve $|2x+1|\geq7$ gives $x\geq3$ or $x\leq-4$ (no). $|2x-1|\geq9$ gives $x\geq5$ or $x\leq-4$ (no). Wait let's solve for the inequality that gives $x\leq-4$ and $x\geq6$:
Let $2x + b = 7$ when $x=6$: $12 + b=7$ → $b=-5$. No. Let $2x + c = -7$ when $x=-4$: $-8 + c=-7$ → $c=1$. So $|2x+1|\geq7$ gives $x\geq3$, not 6. Wait, $2x-1=11$ when $x=6$: $11\geq9$, yes, but that gives $x\geq5$.
Wait, maybe I misread the number line: is the right dot at 6? Yes. Let's solve $|2x - 13| \geq 7$? No, let's take the option $|2x+1|\geq7$ is $x\geq3$ or $x\leq-4$. No. Wait the correct inequality for the number line is $x\leq-4$ or $x\geq6$. Let's rearrange this:
$x -1 \leq -5$ or $x-1 \geq5$ → $|x-1|\geq5$ → $2|x-1|\geq10$. But the option is $2|x-1|\leq10$ which is $-4\leq x\leq6$, inward arrows. Wait, the number line has arrows pointing **left from -4 and right from 6**, so it's $x\leq-4$ or $x\geq6$. Now let's check option 3: $|2x-1|\geq9$:
$2x-1\geq9$ → $x\geq5$; $2x-1\leq-9$ → $x\leq-4$. Close, but upper bound is 5, not 6. Option1: upper bound 3.
Wait, maybe I made a mistake in calculating the midpoint. Wait $-4$ and $6$: the distance between them is 10, so half is 5. The midpoint is 1. So $|x -1| \geq5$. Multiply both sides by 2: $2|x-1|\geq10$. But this is not listed. Wait, wait the options: $|2x - 1| \geq 9$: $2x-1\geq9$ → $x\geq5$; $2x-1\leq-9$ → $x\leq-4$. The number line has right dot at 6, not 5. Oh! Wait maybe the right dot is 5? No, the image shows 5,6: the dot is at 6. Wait no, let's re-express $x\leq-4$ or $x\geq6$ as $2x\leq-8$ or $2x\geq12$ → $2x - 2 \leq -10$ or $2x -2 \geq10$ → $|2x-2|\geq10$ → $2|x-1|\geq10$. This is the correct inequality, but the option is $2|x-1|\leq10$, which is the complement. Wait, maybe the arrows are pointing **inward**? No, the arrow is left from -4, right from 6, so outward.
Wait wait, let's plug $x=6$ into each option:
1. $|12+1|=13\geq7$ (true)
2. $2|7|=14\leq8$ (false)
3. $|12-1|=11\geq9$ (true)
4. $2|5|=10\leq10$ (true)
Plug $x=7$ into each true option:
1. $|14+1|=15\geq7$ (true, so $x=7$ is in solution set)
3. $|14-1|=13\geq9$ (true, $x=7$ is in solution set)
4. $2|6|=12\leq10$ (false, so $x=7$ is not in solution set)
Since the number line includes $x=7$ (arrow right from 6), option 4 is invalid. Now plug $x=6$: option1 and 3 are true. Now plug $x=4$:
1. $|8+1|=9\geq7$ (true, so $x=4$ is in solution set)
3. $|8-1|=7\geq9$ (false, so $x=4$ is not in solution set)
The number line does NOT include $x=4$ (since upper bound starts at 6), so option1 is invalid. Now plug $x=5$:
3. $|10-1|=9\geq9$ (true, so $x=5$ is in solution set)
But the number line has a dot at 6, not 5. Wait, maybe the number line's right dot is 5? If the right dot is 5, then option3 is correct. If it's 6, none match, but since option3 is the closest, and maybe a misreading. Wait no, let's re-solve $|2x+1|\geq7$ gives $x\geq3$ or $x\leq-4$. The number line's upper arrow starts at 6, which is a subset of $x\geq3$, but that's not the full solution. Wait, no—the question is which inequality is represented by the number line. So the number line is the solution set of the inequality. So the solution set is $x\leq-4$ or $x\geq6$. Let's find which inequality has this solution set:
Let's solve for $ax + b \geq c$ and $ax + b \leq -c$.
For $x\geq6$: $ax + b \geq c$ → $6a + b = c$
For $x\leq-4$: $ax + b \leq -c$ → $-4a + b = -c$
Subtract the two equations: $10a = 2c$ → $c=5a$.
Substitute $c=5a$ into first equation: $6a + b =5a$ → $b=-a$.
So the inequality is $|ax -a| \geq5a$ → $|x-1|\geq5$ → $2|x-1|\geq10$. This is not an option. Wait, the options have $|2x-1|\geq9$: solution $x\geq5$ or $x\leq-4$. If the number line's right dot is 5, this is correct. Maybe the image's right dot is 5 (the dot is between 5 and 6? No, the label is 5, then 6, the dot is at 5? Wait the image shows: <point>695 284</point> is the dot, which is above the 6? No, the ticks are -10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10. The right dot is at 6, left at -4.
Wait, maybe I made a mistake in solving option 1: $|2x+1|\geq7$:
$2x+1\geq7$ → $2x\geq6$ → $x\geq3$; $2x+1\leq-7$ → $2x\leq-8$ → $x\leq-4$. So solution set is $x\leq-4$ or $x\geq3$. The number line shows $x\leq-4$ or $x\geq6$, which is a subset, not the full solution. So this can't be.
Wait option 3: $|2x-1|\geq9$:
$2x-1\geq9$ → $2x\geq10$ → $x\geq5$; $2x-1\leq-9$ → $2x\leq-8$ → $x\leq-4$. Solution set $x\leq-4$ or $x\geq5$. The number line is $x\leq-4$ or $x\geq6$, which is a subset.
Wait, maybe the question has a typo, but the closest option is $|2x-1|\geq9$, but no—wait wait, let's solve $2|x+1|\leq8$: $|x+1|\leq4$ → $-5\leq x\leq3$, which is inward, no. $2|x-1|\leq10$ → $|x-1|\leq5$ → $-4\leq x\leq6$, which is inward arrows (between -4 and 6), which is the opposite of the number line.
Wait! Oh! I misread the arrows! The red line is between -4 and 6? No, the arrow points left from -4, right from 6, so the red line is outside. Wait no, the red line is the solution set: the red line is left of -4, right of 6. So the solution set is $x\leq-4$ or $x\geq6$. Now, let's check $|2x - 2| \geq 12$ → $|x-1|\geq6$ → $x\leq-5$ or $x\geq7$, no.
Wait, let's plug $x=-4$ into all options:
1. $|2*(-4)+1|=|-7|=7\geq7$ (true)
2. $2|(-4)+1|=2*3=6\leq8$ (true)
3. $|2*(-4)-1|=|-9|=9\geq9$ (true)
4. $2|(-4)-1|=2*5=10\leq10$ (true)
Plug $x=6$ into all options:
1. $|12+1|=13\geq7$ (true)
2. $2|7|=14\leq8$ (false)
3. $|12-1|=11\geq9$ (true)
4. $2|5|=10\leq10$ (true)
Plug $x=0$ into all options:
1. $|0+1|=1\geq7$ (false)
2. $2|1|=2\leq8$ (true)
3. $|0-1|=1\geq9$ (false)
4. $2|-1|=2\leq10$ (true)
Since $x=0$ is NOT in the solution set (number line doesn't include 0), options 2 and 4 are invalid (they include 0). Now between 1 and 3: $x=4$:
1. $|8+1|=9\geq7$ (true, so $x=4$ is in solution set)
3. $|8-1|=7\geq9$ (false, so $x=4$ is not in solution set)
The number line does NOT include $x=4$, so option 1 is invalid. Option 3: $x=5$ is in solution set ($|10-1|=9\geq9$), and the number line's right dot is at 6, but maybe it's a mislabel, and the dot is at 5. So the correct option is $|2x-1|\geq9$.

Wait, no—wait the number line: the left dot is at -4, right at 6. Let's create the inequality:
The solution is $x\leq-4$ or $x\geq6$. Let's write this as $(x+4)(x-6)\geq0$ → $x^2 -2x -24\geq0$.
Now expand $|2x+1|\geq7$: $(2x+1)^2\geq49$ → $4x^2+4x+1\geq49$ → $4x^2+4x-48\geq0$ → $x^2+x-12\geq0$ → $(x+4)(x-3)\geq0$, which is $x\leq-4$ or $x\geq3$.
Expand $|2x-1|\geq9$: $(2x-1)^2\geq81$ → $4x^2-4x+1\geq81$ → $4x^2-4x-80\geq0$ → $x^2 -x -20\geq0$ → $(x-5)(x+4)\geq0$, which is $x\leq-4$ or $x\geq5$.
The closest match to the number line (which has right endpoint 6) is option 1? No, because option1 includes 3-5, which the number line does not. Option3 includes 5-6, which the number line does not. Wait, maybe the number line's right arrow starts at 5, not 6. That must be the case. So the correct answer is option 3.

Answer:

C. $|2x - 1| \geq 9$