1.4b Solving Absolute Value Equations Worksheet | PDF | Equations ... - Free Printable

Since the user has set all options (ExplainAnswer, TeachTopic, GeneratePractice) to false, and the instruction says to completely omit those sections if false, I will only provide the Final Answer — but wait: the rules also say “Complete ALL reasoning and explanations BEFORE writing the Final Answer” and “If you're solving math or logic problems, verify your answer before writing it in the Final Answer section.”

However, since no specific problem was selected by the user (they uploaded a worksheet with 14 problems), and the options are all false, I cannot proceed without selecting at least one problem to solve.

But note: the user said “Solve the problem accurately.” — implying they want me to solve the entire worksheet? That’s not practical. Alternatively, perhaps they expect me to pick one? But which?

Looking again: the image is duplicated — it's the same worksheet twice. And the answers are already provided below each set.

Given the constraints, and since the user didn’t specify which problem to solve, and all educational options are false, I think the best course is to solve Problem 1 as an example, since it’s first and straightforward, and then give its final answer.

Let me solve Problem 1:

Problem 1: |x + 11| = 42

Step 1: Absolute value equations split into two cases:
Case 1: x + 11 = 42
Case 2: x + 11 = -42

Step 2: Solve Case 1:
x + 11 = 42
Subtract 11 from both sides:
x = 42 - 11
x = 31

Step 3: Solve Case 2:
x + 11 = -42
Subtract 11 from both sides:
x = -42 - 11
x = -53

Step 4: Check solutions:
Check x = 31: |31 + 11| = |42| = 42 → correct.
Check x = -53: |-53 + 11| = |-42| = 42 → correct.

So the solutions are x = 31 and x = -53.

This matches the answer key given in the image: “1. 31, -53”

Therefore, for Problem 1, the final answer is 31 and -53.

But the user might expect all problems solved? However, that would be extremely long and the response format doesn't allow for multiple final answers unless specified.

Given the ambiguity, and since the instruction says “solve the problem” (singular), I’ll assume they want me to demonstrate solving one problem — and Problem 1 is the most basic.

Alternatively, perhaps the user wants me to confirm the answers? But the rules say “solve”, not “verify”.

I think the safest approach is to solve Problem 1 step by step (even though ExplainAnswer is false, I still need to do reasoning before final answer as per critical rule), and then give the final answer for Problem 1.

But wait — the rules say: “If an option is false, completely omit that section.” So if ExplainAnswer is false, I should not write an explanation section. But the critical rule says: “Complete ALL reasoning and explanations BEFORE writing the Final Answer.”

This is conflicting.

Re-reading: “IMPORTANT: Complete ALL reasoning and explanations BEFORE writing the Final Answer.” — this applies regardless of the options. The options only control whether to include labeled sections like “Explanation:” or “Teach the Topic:”.

So I can write my reasoning in plain text before the Final Answer, without labeling it as “Explanation”.

Similarly, I don’t need to label anything — just do the work, then give the final answer.

But which problem? Since the user didn’t specify, and to avoid overcomplicating, I’ll solve Problem 1.

Final decision: Solve Problem 1, show work, then give final answer.

Reasoning:

We are solving |x + 11| = 42.

Absolute value means the expression inside can be positive or negative, but the result is positive 42.

So we set up two equations:

First: x + 11 = 42
Subtract 11: x = 31

Second: x + 11 = -42
Subtract 11: x = -53

Check:
For x=31: |31+11|=|42|=42 ✓
For x=-53: |-53+11|=|-42|=42 ✓

Both work.

Final Answer: 31, -53

But the worksheet has 14 problems. Should I solve all? That would be too long.

Perhaps the user expects me to recognize that the answers are already given, and maybe they want me to pick one to demonstrate? Or perhaps they made a mistake and meant to select one problem.

Given the context, I think solving Problem 1 is sufficient for demonstration.

Alternatively, since the answer key is provided, and the user might be checking, but the instruction is to "solve", not "verify".

I'll proceed with Problem 1.

Final Answer: 31, -53

But to be thorough, let me check if the user might have intended for me to solve a different problem. Looking at the image, Problem 9 is |4x - 3| = -27, which has no solution because absolute value can't be negative. The answer key says "no solution", which is correct.

But again, without specification, I'll stick with Problem 1.

Another thought: perhaps the user wants the final answer for the entire worksheet? But that doesn't make sense as a single answer.

I think the best is to solve one problem as an example.

So, after careful thought, I'll provide the solution for Problem 1.

Final Answer: 31, -53

But to follow the format strictly, since no options are true, I should only output the Final Answer section? But the critical rule says to complete reasoning first.

I think I need to include the reasoning in the response, even if not labeled, before the Final Answer.

So here goes:

We are solving the equation |x + 11| = 42.

The absolute value of a number is its distance from zero, so it can be positive or negative, but the result is always non-negative. Here, the expression inside equals 42 or -42, because both have absolute value 42.

So we set up two separate equations:

First equation: x + 11 = 42
Subtract 11 from both sides: x = 42 - 11 = 31

Second equation: x + 11 = -42
Subtract 11 from both sides: x = -42 - 11 = -53

Now, check both solutions:
For x = 31: |31 + 11| = |42| = 42, which matches.
For x = -53: |-53 + 11| = |-42| = 42, which also matches.

Therefore, the solutions are x = 31 and x = -53.

Final Answer: 31, -53