Graphing Inequalities worksheet with number lines and algebraic expressions.

To solve these problems, we need to match each inequality statement (like $b \le -2.5$) with the correct number line graph.

Here is how to read the graphs:
1. The Dot:
* A solid (filled-in) dot means the number is included. This matches symbols $\le$ (less than or equal to) and $\ge$ (greater than or equal to).
* An open (empty) circle means the number is *not* included. This matches symbols $<$ (less than) and $>$ (greater than).
2. The Arrow:
* An arrow pointing to the left means "less than" ($<$ or $\le$).
* An arrow pointing to the right means "greater than" ($>$ or $\ge$).

Let's go through them one by one:

1) $b \le -2.5$
* Symbol: $\le$ means solid dot and arrow to the left.
* Number: $-2.5$ is halfway between $-2$ and $-3$.
* Look for a graph with a solid dot at $-2.5$ shading left.
* Match: Graph 10) has a solid dot at $-2.5$ and shades left.

2) $z$ (Looking at graph 2)
* Graph 2 shows an open circle at $-2$ shading left.
* This represents $z < -2$.
* Let's check the list for $z < -2$. It's not explicitly listed as a question on the left, but let's look at the other side. Wait, the worksheet asks us to fill in the blank line next to the number. The questions are numbered 1-20.
* Questions 1, 3, 5, 6, 7, 9, 11, 13, 14, 16, 17, 18, 19, 20 provide the inequality. We need to find the matching graph letter/number? No, looking closely at the layout:
* Left column (1-10): Provides an inequality equation. There is a blank line. The graphs below are labeled with letters ($b, z, f, x, n, d, w, c$ etc are variables, but the graphs themselves aren't labeled A, B, C... wait. The graphs *are* the answers? Or are we matching?)
* Actually, looking at items 11-20, they show a graph and ask for the inequality. Items 1-10 show an inequality and have a blank line where a graph should be drawn? No, the graphs are already printed below each question.
* Let's re-read the sheet structure.
* Item 1: Inequality $b \le -2.5$. Below it is a number line. Is that the answer key? No, the red ink suggests these are completed or example problems.
* Actually, usually these worksheets work like this:
* Odd numbers might be "Write the inequality for the graph" and Even numbers "Graph the inequality".
* Let's look at #11. It shows a graph and a blank line for the inequality.
* Let's look at #1. It shows an inequality and a graph below it. The graph matches the inequality perfectly ($b \le -2.5$ corresponds to the graph shown in #1? No, the graph in #1 is just a number line. The red drawing is ON the number line).
* Ah, I see. The red markings are the solutions. The user wants me to verify or provide the answers for the blanks.
* Let's look at #11. The graph has an open circle at 9 and shades left. The blank asks for the inequality.
* Open circle at 9 means $< 9$ or $> 9$.
* Shading left means $<$.
* Variable is $h$.
* Answer: $h < 9$.
* Let's look at #12. Graph has open circle at 12, shades left. Variable $w$.
* Answer: $w < 12$.
* Let's look at #13. Inequality: $-d \ge 2$.
* To graph this, we usually isolate $d$. Divide by $-1$ and flip the sign: $d \le -2$.
* So we need a solid dot at $-2$ shading left.
* The graph shown in #13 is empty. We need to describe the graph or solve it? The prompt says "Solve the problem". The problems are mixed. Some give the inequality, some give the graph.
* Let's assume the task is to fill in the missing information.
* For 1-10, the inequality is given. The graph is drawn in red. It seems the red drawings ARE the answers provided by a student or teacher.
* For 11-20, some have graphs and need inequalities, some have inequalities and need graphs.
* Let's look at #11 again. The graph is drawn in red. The inequality line is blank. So the task is to write the inequality.
* Let's look at #13. The inequality is given. The graph area is blank. So the task is to draw the graph? But I cannot draw. I must describe it or provide the final mathematical answer.
* Let's look at #14. Graph is drawn (open circle at -6, shade right). Inequality line is blank. Task: Write inequality. Variable $w$.
* Open circle at -6, shade right $\rightarrow w > -6$.
* Let's look at #15. Graph is drawn (open circle at -4, shade right). Inequality line is blank. Variable $b$.
* Answer: $b > -4$.
* Let's look at #16. Inequality $-k < 3$. Graph is blank.
* Solve for $k$: Divide by $-1$, flip sign $\rightarrow k > -3$.
* Graph description: Open circle at $-3$, shade right.
* Let's look at #17. Inequality $-z \ge 2.5$. Graph is blank.
* Solve for $z$: Divide by $-1$, flip sign $\rightarrow z \le -2.5$.
* Graph description: Solid dot at $-2.5$, shade left.
* Let's look at #18. Inequality $12 > c$. Graph is blank.
* Rewrite as $c < 12$.
* Graph description: Open circle at $12$, shade left.
* Let's look at #19. Inequality $d < 5$. Graph is blank.
* Graph description: Open circle at $5$, shade left.
* Let's look at #20. Inequality $-r \le 0$. Graph is blank.
* Solve for $r$: Divide by $-1$, flip sign $\rightarrow r \ge 0$.
* Graph description: Solid dot at $0$, shade right.

* Now let's re-evaluate 1-10. They have inequalities and RED GRAPHS. Are the red graphs correct?
* 1) $b \le -2.5$. Graph: Solid dot at -2.5, shade left. Correct.
* 2) Just variable $z$? No, look at the graph. Open circle at -2, shade left. The text above is just "$z$". This looks like a typo in my reading or the sheet. Let's look closer. Ah, item 2 has "$z$" and a blank line? No, it looks like the inequality is missing or it's asking to interpret the graph. But wait, item 1 has an inequality. Item 2 has a graph. Item 3 has an inequality. Item 4 has a graph?
* Let's trace carefully.
* 1) Inequality given. Graph drawn.
* 2) Variable $z$ given. Graph drawn (Open circle -2, left). This implies the question was "Graph $z < -2$" but the inequality part is missing/cut off? Or maybe the student filled it in?
* Actually, looking at the pattern:
* 1, 3, 5, 6, 7, 9 are inequalities.
* 2, 4, 8, 10 are... wait.
* Let's look at 2. The text is just "$z$". The graph is open circle at -2, left. This corresponds to $z < -2$.
* Let's look at 4. Text "$x$". Graph open circle at 2, left. Corresponds to $x < 2$.
* Let's look at 8. Text "$w$". Graph solid dot at 9, left. Corresponds to $w \le 9$.
* Let's look at 10. Text "$c$". Graph solid dot at -2.5, left. Corresponds to $c \le -2.5$.
* It seems for 2, 4, 8, 10, the inequality symbol and number were erased or missing, and the graph is the answer key?
* OR, more likely, this is a worksheet where the student has to fill in the blanks. The red ink is the "Answer Key" or the "Completed Work".
* The prompt asks me to "Solve the problem". Since the red ink provides the visual solutions for 1-10 and 11-15, I should focus on providing the textual answers for the blanks in 11-20, and verifying/describing the solutions for 1-10 if needed.
* However, usually, when users upload these, they want the answers for the empty blanks.
* Blanks exist at:
* 11) Line for inequality.
* 12) Line for inequality.
* 13) Graph is empty.
* 14) Line for inequality.
* 15) Line for inequality.
* 16) Graph is empty.
* 17) Graph is empty.
* 18) Graph is empty.
* 19) Graph is empty.
* 20) Graph is empty.

* What about 1-10? The lines next to the inequalities are blank in the original print, but the graph below is filled. The "Score" and "Name" fields are blank. The red ink is likely the correction.
* If the user wants me to solve the problems, I will provide the correct inequality for the graphs shown in 11-15, and the correct graph description for the inequalities in 13, 16-20. For 1-10, since the graphs are already drawn correctly in red, I will confirm they are correct or simply list the solution if it's considered a "problem" to verify.
* Let's assume the standard request: Provide the missing information for all 20 items.

Step-by-Step Derivation:

Item 1: $b \le -2.5$
* Solution: Solid dot at $-2.5$, arrow to the left.
* (Graph shown is correct).

Item 2: The text is just "$z$". The graph shows open circle at $-2$, arrow left.
* Inferred Problem: Graph $z < -2$.
* Solution: Open circle at $-2$, arrow to the left.
* (Graph shown is correct for $z < -2$).

Item 3: $f$ ... wait, looking at crop 1, item 3 is $f$ with a solid dot at 5 shading right? No, the red dot is at 5. The inequality is missing?
* Let's look at the original full image again.
* 1) $b \le -2.5$
* 2) $z$ (Graph: $< -2$) -> Likely meant to be $z < -2$
* 3) $f$ (Graph: Solid dot at 5, shade right) -> Likely meant to be $f \ge 5$
* 4) $x$ (Graph: Open circle at 2, shade left) -> Likely meant to be $x < 2$
* 5) $n \le 13$ (Graph: Solid dot at 13, shade left) -> Correct.
* 6) $-d \le 2.5$ -> Solve: $d \ge -2.5$. Graph: Solid dot at -2.5, shade right.
* Wait, look at graph 6. It's empty in the black ink, but there is no red ink on graph 6?
* Let's look really closely at the full image.
* Red ink is on: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15.
* Red ink is NOT on: 6, 13, 16, 17, 18, 19, 20.
* Okay, so 6, 13, 16-20 are the unsolved ones?
* And 11, 12, 14, 15 have graphs but need inequalities written.
* And 1-5, 7-10 are solved examples?
* This is confusing. Let's look at the structure again.
* Maybe the red ink IS the student's work, and I need to grade it? Or provide the correct answers for all?
* "Solve the problem accurately." implies I should provide the correct answer for every item numbered 1-20.

Let's produce the correct answer for each number 1-20.

1) $b \le -2.5$
* Graph: Closed circle at $-2.5$, shade left.

2) $z$ (Context from graph: Open circle at -2, shade left)
* Inequality: $z < -2$

3) $f$ (Context from graph: Closed circle at 5, shade right)
* Inequality: $f \ge 5$

4) $x$ (Context from graph: Open circle at 2, shade left)
* Inequality: $x < 2$

5) $n \le 13$
* Graph: Closed circle at $13$, shade left.

6) $-d \le 2.5$
* Step 1: Divide by $-1$ and flip the inequality sign.
* $d \ge -2.5$
* Graph: Closed circle at $-2.5$, shade right.

7) $2 \ge w$
* Rewrite: $w \le 2$
* Graph: Closed circle at $2$, shade left.
* (Note: The red graph in the image shows closed circle at 2, shade left. This is correct).

8) $w$ (Context from graph: Closed circle at 9, shade left)
* Inequality: $w \le 9$

9) $d$ (Context from graph: Closed circle at 0, shade right?? No, look at #9. Red dot at 0. Arrow points RIGHT. Wait.
* Let's look at #9 in the image.
* Text: "$d$".
* Graph: Red solid dot at 0. Red arrow to the RIGHT.
* Inequality: $d \ge 0$.

10) $c$ (Context from graph: Solid dot at -2.5, shade left)
* Inequality: $c \le -2.5$

11) Graph: Open circle at 9, shade left. Variable $h$.
* Inequality: $h < 9$

12) Graph: Open circle at 12, shade left. Variable $w$.
* Inequality: $w < 12$

13) $-d \ge 2$
* Step 1: Divide by $-1$ and flip sign.
* $d \le -2$
* Graph: Closed circle at $-2$, shade left.

14) Graph: Open circle at -6, shade right. Variable $w$.
* Inequality: $w > -6$

15) Graph: Open circle at -4, shade right. Variable $b$.
* Inequality: $b > -4$

16) $-k < 3$
* Step 1: Divide by $-1$ and flip sign.
* $k > -3$
* Graph: Open circle at $-3$, shade right.

17) $-z \ge 2.5$
* Step 1: Divide by $-1$ and flip sign.
* $z \le -2.5$
* Graph: Closed circle at $-2.5$, shade left.

18) $12 > c$
* Rewrite: $c < 12$
* Graph: Open circle at $12$, shade left.

19) $d < 5$
* Graph: Open circle at $5$, shade left.

20) $-r \le 0$
* Step 1: Divide by $-1$ and flip sign.
* $r \ge 0$
* Graph: Closed circle at $0$, shade right.

Summary of Output:
I will provide the final answer as a list corresponding to numbers 1-20. For items requiring a graph, I will describe the graph clearly (Dot type, Position, Direction). For items requiring an inequality, I will write the inequality.

Let's double check calculations.
6) $-d \le 2.5 \rightarrow d \ge -2.5$. Correct.
13) $-d \ge 2 \rightarrow d \le -2$. Correct.
16) $-k < 3 \rightarrow k > -3$. Correct.
17) $-z \ge 2.5 \rightarrow z \le -2.5$. Correct.
20) $-r \le 0 \rightarrow r \ge 0$. Correct.

Check graph readings:
11) Open circle 9, left. $h < 9$. Correct.
12) Open circle 12, left. $w < 12$. Correct.
14) Open circle -6, right. $w > -6$. Correct.
15) Open circle -4, right. $b > -4$. Correct.

Check inferred inequalities from partial text (2,3,4,8,9,10):
2) Graph is $z < -2$. Text is $z$. Answer: $z < -2$.
3) Graph is $f \ge 5$. Text is $f$. Answer: $f \ge 5$.
4) Graph is $x < 2$. Text is $x$. Answer: $x < 2$.
8) Graph is $w \le 9$. Text is $w$. Answer: $w \le 9$.
9) Graph is $d \ge 0$. Text is $d$. Answer: $d \ge 0$.
10) Graph is $c \le -2.5$. Text is $c$. Answer: $c \le -2.5$.

The request is simple: "Solve the problem". I will list the solutions for 1-20.

Final Answer Format:
1) Graph: Solid dot at -2.5, shade left.
2) Inequality: $z < -2$
3) Inequality: $f \ge 5$
4) Inequality: $x < 2$
5) Graph: Solid dot at 13, shade left.
6) Graph: Solid dot at -2.5, shade right. (Derived from $d \ge -2.5$)
7) Graph: Solid dot at 2, shade left.
8) Inequality: $w \le 9$
9) Inequality: $d \ge 0$
10) Inequality: $c \le -2.5$
11) Inequality: $h < 9$
12) Inequality: $w < 12$
13) Graph: Solid dot at -2, shade left. (Derived from $d \le -2$)
14) Inequality: $w > -6$
15) Inequality: $b > -4$
16) Graph: Open circle at -3, shade right. (Derived from $k > -3$)
17) Graph: Solid dot at -2.5, shade left. (Derived from $z \le -2.5$)
18) Graph: Open circle at 12, shade left. (Derived from $c < 12$)
19) Graph: Open circle at 5, shade left.
20) Graph: Solid dot at 0, shade right. (Derived from $r \ge 0$)

I will group them or list them clearly. Since the user didn't specify format, a numbered list is best. I will clarify whether the answer is an inequality or a graph description.

Wait, for #1, 5, 7, the inequality is given and the graph is the answer.
For #11, 12, 14, 15, the graph is given and the inequality is the answer.
For #6, 13, 16-20, the inequality is given and the graph is the answer.
For #2, 3, 4, 8, 9, 10, the variable is given, and the graph is drawn, implying the inequality is the missing piece of info to make sense of the graph, OR the inequality was cut off. Given the pattern of the other questions, it's highly likely these were meant to be "Write the inequality for the graph" or "Graph the inequality" where the text was incomplete. I will provide the inequality that matches the graph shown for these items, as that completes the pair.

Let's refine the final output to be very clear for a student.

Final Answer Construction:
1. Graph: Place a solid dot at -2.5 and shade to the left.
2. Inequality: $z < -2$
3. Inequality: $f \ge 5$
4. Inequality: $x < 2$
5. Graph: Place a solid dot at 13 and shade to the left.
6. Graph: First solve $-d \le 2.5$ to get $d \ge -2.5$. Place a solid dot at -2.5 and shade to the right.
7. Graph: Rewrite $2 \ge w$ as $w \le 2$. Place a solid dot at 2 and shade to the left.
8. Inequality: $w \le 9$
9. Inequality: $d \ge 0$
10. Inequality: $c \le -2.5$
11. Inequality: $h < 9$
12. Inequality: $w < 12$
13. Graph: First solve $-d \ge 2$ to get $d \le -2$. Place a solid dot at -2 and shade to the left.
14. Inequality: $w > -6$
15. Inequality: $b > -4$
16. Graph: First solve $-k < 3$ to get $k > -3$. Place an open circle at -3 and shade to the right.
17. Graph: First solve $-z \ge 2.5$ to get $z \le -2.5$. Place a solid dot at -2.5 and shade to the left.
18. Graph: Rewrite $12 > c$ as $c < 12$. Place an open circle at 12 and shade to the left.
19. Graph: Place an open circle at 5 and shade to the left.
20. Graph: First solve $-r \le 0$ to get $r \ge 0$. Place a solid dot at 0 and shade to the right.