Question

use the image to answer the question.
preston solved an inequality and graphed the solution on the number line. which of the following inequalities did he solve?
(1 point)
\\( 0.9x + 2.8 < 5.95 \\)
\\( 5.6x - 18.9 \leq 0.7 \\)
\\( 2.1x - 5.6 \geq 8.75 \\)
\\( 3.4x - 1.2 > 4.75 \\)

Explanation:

Step1: Analyze the number line

The graph has a closed dot at 1 and points right, so the solution is $x \geq 1$.

Step2: Solve Option 1

Isolate $x$ for $0.9x + 2.8 \leq 5.95$:
$0.9x \leq 5.95 - 2.8$
$0.9x \leq 3.15$
$x \leq \frac{3.15}{0.9} = 3.5$
This does not match $x \geq 1$.

Step3: Solve Option 2

Isolate $x$ for $5.6x - 18.9 \leq 0.7$:
$5.6x \leq 0.7 + 18.9$
$5.6x \leq 19.6$
$x \leq \frac{19.6}{5.6} = 3.5$
This does not match $x \geq 1$.

Step4: Solve Option 3

Isolate $x$ for $2.1x - 5.6 \geq 8.75$:
$2.1x \geq 8.75 + 5.6$
$2.1x \geq 14.35$
$x \geq \frac{14.35}{2.1} \approx 6.83$
This does not match $x \geq 1$.

Step5: Solve Option 4

Isolate $x$ for $3.4x - 1.2 \geq 4.75$:
$3.4x \geq 4.75 + 1.2$
$3.4x \geq 5.95$
$x \geq \frac{5.95}{3.4} = 1.75$
*Correction: Recheck the number line interpretation and inequalities. Re-express the correct matching:
Wait, re-express the correct inequality that gives $x \geq 1$:
Let's solve $3.4x - 1.2 \geq 2.2$ (no, use given options):
Wait, correct calculation for Option 2 was wrong: $5.6x - 18.9 \leq 0.7$ gives $x \leq 3.5$ (left of 3.5, closed dot).
Wait, the number line is $x \geq 1$. Let's find which inequality gives $x \geq 1$:
Suppose we solve for $x \geq 1$:
Test $x=1$ in each inequality:
1. $0.9(1)+2.8=3.7 \leq 5.95$ true, but solution is $x \leq 3.5$
2. $5.6(1)-18.9=-13.3 \leq 0.7$ true, solution $x \leq 3.5$
3. $2.1(1)-5.6=-3.5 \geq 8.75$ false
4. $3.4(1)-1.2=2.2 \geq 4.75$ false
Wait, re-interpret the number line: closed dot at 1, arrow right is $x \geq 1$. Let's solve each inequality correctly:

Step1: Solve $0.9x + 2.8 \leq 5.95$

$0.9x \leq 3.15$
$x \leq 3.5$

Step2: Solve $5.6x - 18.9 \leq 0.7$

$5.6x \leq 19.6$
$x \leq 3.5$

Step3: Solve $2.1x - 5.6 \geq 8.75$

$2.1x \geq 14.35$
$x \geq 6.83$

Step4: Solve $3.4x - 1.2 \geq 4.75$

$3.4x \geq 5.95$
$x \geq 1.75$
Wait, there is a discrepancy. Wait, maybe the number line is $x \geq 1$, so let's find which inequality has solution $x \geq 1$:
Let's reverse: if $x \geq 1$, multiply by 3.4: $3.4x \geq 3.4$, then $3.4x -1.2 \geq 3.4-1.2=2.2$, but the option is $3.4x-1.2 \geq4.75$.
Wait, maybe I misread the number line: closed dot at 1, arrow right is $x \geq 1$. Let's check the option $5.6x - 18.9 \geq 0.7$ (maybe a typo in the image, $\leq$ vs $\geq$):
If it's $5.6x -18.9 \geq 0.7$:
$5.6x \geq 19.6$
$x \geq 3.5$ no.
Wait, $0.9x -2.8 \geq -1.9$: $0.9x \geq 0.9$, $x \geq1$. But that's not an option.
Wait, recheck the original image: the selected option is $5.6x -18.9 \leq0.7$, but its solution is $x \leq3.5$, which would be a closed dot at 3.5, arrow left.
Wait, maybe the number line is closed dot at 1, arrow left? No, arrow is right.
Wait, correct calculation for $3.4x -1.2 \geq4.75$:
$3.4x=4.75+1.2=5.95$
$x=5.95/3.4=1.75$, so $x \geq1.75$, which is close to 1, but not 1.
Wait, $2.1x -5.6 \geq -3.5$: $2.1x \geq2.1$, $x \geq1$, but the option is $2.1x-5.6 \geq8.75$.
Wait, maybe the number line is $x \leq1$? Closed dot at 1, arrow left. Then solve $0.9x+2.8 \leq3.7$: $0.9x \leq0.9$, $x \leq1$. But the option is $0.9x+2.8 \leq5.95$.
Wait, $5.6x -18.9 \leq-13.3$: $5.6x \leq5.6$, $x \leq1$. Which is $5.6x -18.9 \leq-13.3$, but the option is $5.6x-18.9 \leq0.7$.
Ah! I see, the number line in the image has a closed dot at 1, arrow right, so $x \geq1$. The only inequality whose solution includes $x \geq1$ and matches is:
Wait, no, let's test $x=1$ in each inequality:
1. $0.9(1)+2.8=3.7 \leq5.95$: true, solution $x \leq3.5$ (includes $x \geq1$ but is not the full solution)
2. $5.6(1)-18.9=-13.3 \leq0.7$: true, solution $x \leq3.5$ (includes $x \geq1$ but is not the full solution)
3. $2.1(1)-5.6=-3.5 \geq8.75$: false
4. $3.4(1)-1.2=2.2 \geq4.75$: false
Wait, the image's selected option is $5.6x -18.9 \leq0.7$, which has solution $x \leq3.5$. That would be a closed dot at 3.5, arrow left. So maybe the number line was misread: closed dot at 3.5, arrow left? No, the dot is at 1.
Wait, correct the number line: closed dot at 1, arrow right is $x \geq1$. The inequality that gives this is:
Let's solve for $x \geq1$:
Multiply by 0.9: $0.9x \geq0.9$, add 2.8: $0.9x+2.8 \geq3.7$, not the option.
Multiply by 5.6: $5.6x \geq5.6$, add 18.9: $5.6x+18.9 \geq24.5$, no.
Multiply by 2.1: $2.1x \geq2.1$, add 5.6: $2.1x+5.6 \geq7.7$, no.
Multiply by 3.4: $3.4x \geq3.4$, add 1.2: $3.4x+1.2 \geq4.6$, no.
Wait, maybe the inequality is $5.6x - 18.9 \geq -13.3$, which is $5.6x \geq5.6$, $x \geq1$. Which is equivalent to $5.6x -18.9 \geq -13.3$, but the option is $5.6x-18.9 \leq0.7$.
Wait, maybe the image has a typo, but the selected option is $5.6x -18.9 \leq0.7$, which has solution $x \leq3.5$. If the number line was closed dot at 3.5, arrow left, that's correct. But the number line shows dot at 1.
Wait, re-calculate $5.6x -18.9 \leq0.7$:
$5.6x \leq0.7+18.9=19.6$
$x \leq19.6/5.6=3.5$. Correct.
$0.9x+2.8 \leq5.95$: $x \leq3.5$, same solution.
$2.1x-5.6 \geq8.75$: $x \geq14.35/2.1≈6.83$
$3.4x-1.2 \geq4.75$: $x \geq5.95/3.4=1.75$
The number line shows $x \geq1$, so the closest is $3.4x-1.2 \geq4.75$ with $x \geq1.75$, which is a subset of $x \geq1$. But the only inequality that could match if the number line is $x \geq1.75$ is $3.4x-1.2 \geq4.75$.
Wait, maybe the dot is at 1.75, not 1. That would make sense. So assuming the dot is at 1.75, the solution is $x \geq1.75$, which is the solution to $3.4x -1.2 \geq4.75$.

But if we go by the number line as $x \geq1$, the correct inequality is not listed, but the only one with solution that includes $x \geq1$ and is a left-bound is none. Wait, no, the options:
Wait, I made a mistake in Step 2: $5.6x -18.9 \leq0.7$ gives $x \leq3.5$, which is a closed dot at 3.5, arrow left. That would match a number line with dot at 3.5, arrow left.
The number line in the image has dot at 1, arrow right, so $x \geq1$. The only inequality that gives $x \geq1$ is if we have $3.4x -1.2 \geq2.2$, which is not an option.
Wait, maybe the question is asking which inequality's solution is graphed, and the graph is $x \geq1$. Let's solve each inequality:
1. $0.9x +2.8 \leq5.95$: $x \leq3.5$
2. $5.6x -18.9 \leq0.7$: $x \leq3.5$
3. $2.1x -5.6 \geq8.75$: $x \geq6.83$
4. $3.4x -1.2 \geq4.75$: $x \geq1.75$
The closest to $x \geq1$ is $3.4x -1.2 \geq4.75$ with $x \geq1.75$, which is a right-bound inequality, matching the arrow direction.

Answer:

D. $3.4x - 1.2 \geq 4.75$

Note: If the number line is correctly read as $x \geq1$, there is a discrepancy, but based on the arrow direction (right, so $\geq$ inequality), the only $\geq$ inequality with solution closest to the dot at 1 is $3.4x -1.2 \geq4.75$ with $x \geq1.75$, likely a misreading of the dot position as 1 instead of 1.75.