Math 3 Quiz: Solving Absolute Value Equations

9 questions · exam conditions

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Solving Absolute Value EquationsQuestion 1 of 9

A student attempts to solve $|2x - 6| = x + 1$ and obtains potential solutions $x = \frac{7}{3}$ and $x = \frac{5}{3}$ . After substituting back into the original equation, what should the student conclude?

Both solutions are valid since they were derived using correct algebraic methods

Only x = 7/3 is valid; x = 5/3 produces a contradiction when checked

Only x = 5/3 is valid; x = 7/3 produces a contradiction when checked

Both solutions are invalid; the equation has no real solutions

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Math 3 Quiz

Math 3 Quiz: Solving Absolute Value Equations

Practice Solving Absolute Value Equations in Math 3 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solving Absolute Value Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 3.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Both solutions are valid since they were derived using correct algebraic methods
Only x = 7/3 is valid; x = 5/3 produces a contradiction when checked
Only x = 5/3 is valid; x = 7/3 produces a contradiction when checked (correct answer)
Both solutions are invalid; the equation has no real solutions

Explanation: Let's check both potential solutions in |2x - 6| = x + 1. For x = 7/3: |2(7/3) - 6| = |14/3 - 18/3| = |-4/3| = 4/3, and x + 1 = 7/3 + 3/3 = 10/3. Since 4/3 ≠ 10/3, x = 7/3 is extraneous. For x = 5/3: |2(5/3) - 6| = |10/3 - 18/3| = |-8/3| = 8/3, and x + 1 = 5/3 + 3/3 = 8/3. Since 8/3 = 8/3 ✓, x = 5/3 is valid. Therefore, only x = 5/3 is a valid solution.

Question 2

An engineer designing a bridge support system encounters the constraint equation $2|x + 4| - 3 = |x + 4| + 5$ . To determine the valid positions for the support beams, the engineer must solve this equation. What are all possible values of $x$ ?

$x = 4$ and $x = -12$ , providing two optimal beam placement positions (correct answer)
$x = 4$ and $x = -8$ , providing two optimal beam placement positions
$x = -4$ and $x = 12$ , providing two optimal beam placement positions
$x = 8$ and $x = -4$ , providing two optimal beam placement positions

Explanation: Let u = |x + 4|. The equation becomes 2u - 3 = u + 5, which simplifies to u = 8. So |x + 4| = 8, giving x + 4 = 8 or x + 4 = -8, thus x = 4 or x = -12. Choice B incorrectly solves x + 4 = -8 as x = -8. Choice C incorrectly uses x = -4 from setting the expression inside absolute value to zero. Choice D combines errors from other choices.

Question 3

A physics student models the temperature variation in a lab experiment using the equation $|3x - 12| + 7 = 22$ , where $x$ represents time in minutes. If the student needs to identify all times when this specific temperature occurs, which statement about the solutions is correct?

The equation has solutions at $x = 9$ and $x = -1$ , representing two distinct time periods (correct answer)
The equation has solutions at $x = 9$ and $x = 1$ , representing two distinct time periods
The equation has solutions at $x = 7$ and $x = -1$ , representing two distinct time periods
The equation has solutions at $x = 7$ and $x = 1$ , representing two distinct time periods

Explanation: First, isolate the absolute value: |3x - 12| = 15. This gives us two equations: 3x - 12 = 15 or 3x - 12 = -15. Solving: 3x = 27 or 3x = -3, so x = 9 or x = -1. Choice B incorrectly solves 3x - 12 = -15 as x = 1. Choice C uses the wrong isolated value (8 instead of 15). Choice D combines both errors.

Question 4

A meteorologist analyzing storm patterns finds that wind speed data follows the relationship $3|x - 2| + 4 = 2|x - 2| + 9$ . If $x$ represents hours after midnight, at what times does this specific wind condition occur?

At $x = 7$ and $x = -3$ , corresponding to 7:00 AM and 9:00 PM the previous day (correct answer)
At $x = 5$ and $x = -1$ , corresponding to 5:00 AM and 11:00 PM the previous day
At $x = 2$ and $x = 8$ , corresponding to 2:00 AM and 8:00 AM the same day
At $x = 4$ and $x = 0$ , corresponding to 4:00 AM and midnight the same day

Explanation: Let u = |x - 2|. The equation becomes 3u + 4 = 2u + 9, so u = 5. Therefore |x - 2| = 5, giving x - 2 = 5 or x - 2 = -5, so x = 7 or x = -3. Choice B uses u = 3 instead of u = 5. Choice C incorrectly assumes x = 2 is always a solution and makes calculation errors. Choice D uses u = 2 and incorrect arithmetic.

Question 5

A biomedical engineer developing a drug delivery system encounters the dosage optimization equation $|x - 3|^2 = 4$ . The engineer must determine all therapeutic dosage levels that satisfy this constraint for patient safety protocols. What are the valid dosage parameters?

$x = 5$ and $x = 1$ , representing safe therapeutic ranges with different bioavailability profiles (correct answer)
$x = 7$ and $x = -1$ , representing safe therapeutic ranges with different bioavailability profiles
$x = 5$ and $x = -1$ , representing safe therapeutic ranges with different bioavailability profiles
$x = 1$ only, representing a single optimal dosage with no alternative therapeutic options

Explanation: Taking the square root of both sides: |x - 3| = 2 (since we need the positive square root). This gives us x - 3 = 2 or x - 3 = -2, so x = 5 or x = 1. Choice B incorrectly uses |x - 3| = ±2 leading to wrong solutions. Choice C mixes one correct and one incorrect solution. Choice D ignores one of the two valid solutions.

Question 6

If $|3x - 7| + 2 = 11$ , what is the sum of all solutions?

$\frac{14}{3}$ (correct answer)
$\frac{7}{3}$
$7$
$\frac{16}{3}$

Explanation: First isolate the absolute value: |3x - 7| + 2 = 11 → |3x - 7| = 9. This gives us two cases: (1) 3x - 7 = 9 → 3x = 16 → x = 16/3, and (2) 3x - 7 = -9 → 3x = -2 → x = -2/3. The sum of solutions is 16/3 + (-2/3) = 14/3. Choice B (7/3) results from incorrectly solving 3x = 7 instead of 3x = 16. Choice C (7) comes from forgetting to divide by 3. Choice D (16/3) is just one solution, not the sum.

Question 7

A temperature control system maintains a server room using the constraint $|T - 68| < 4$ , where $T$ is temperature in Fahrenheit. If the system logs temperatures of 63.5°F, 71.8°F, and 72.5°F during a monitoring period, what can be concluded?

All recorded temperatures indicate the system is operating within specifications
Two temperatures are within specifications, but one reading indicates a system malfunction (correct answer)
Only one temperature reading shows the system operating correctly
All three temperatures exceed the allowable range, indicating total system failure

Explanation: The constraint |T - 68| < 4 means -4 < T - 68 < 4, so 64 < T < 72. Checking each temperature: 63.5°F is not in (64, 72) since 63.5 < 64 (fails). 71.8°F is in (64, 72) (passes). 72.5°F is not in (64, 72) since 72.5 > 72 (fails). Therefore, two temperatures (63.5°F and 72.5°F) are outside specifications, while one (71.8°F) is within specifications. This indicates two violations and one acceptable reading.

Question 8

When solving $|2x + 1| = |x - 3|$ , a student sets up the equation $2x + 1 = x - 3$ and finds $x = -4$ . What should the student do next?

Check if x = -4 satisfies the original equation, then solve the second case
The solution is complete since one case has been solved correctly
Start over because x = -4 makes both expressions inside negative
Solve 2x + 1 = -(x - 3) next, then check both solutions in the original equation (correct answer)

Explanation: When solving |2x + 1| = |x - 3|, we need to consider both cases: (1) 2x + 1 = x - 3 and (2) 2x + 1 = -(x - 3). The student completed case 1 correctly, getting x = -4. However, they must also solve case 2: 2x + 1 = -(x - 3) → 2x + 1 = -x + 3 → 3x = 2 → x = 2/3. Then both solutions must be checked in the original equation. Choice A suggests checking first, but it's more efficient to find all potential solutions first. Choice B is incorrect because absolute value equations typically have two cases. Choice C is wrong because having negative expressions doesn't invalidate the solution method.

Question 9

The equation $|x - 4| = 2x - 1$ has how many valid solutions?

No solutions exist for this equation
Exactly one solution exists for this equation (correct answer)
Exactly two solutions exist for this equation
Infinitely many solutions exist for this equation

Explanation: We solve by cases. Case 1: x - 4 = 2x - 1 → -x = 3 → x = -3. Case 2: x - 4 = -(2x - 1) → x - 4 = -2x + 1 → 3x = 5 → x = 5/3. Now we must check both solutions in the original equation. For x = -3: |(-3) - 4| = |-7| = 7, and 2(-3) - 1 = -7. Since 7 ≠ -7, x = -3 is extraneous. For x = 5/3: |(5/3) - 4| = |-7/3| = 7/3, and 2(5/3) - 1 = 10/3 - 3/3 = 7/3 ✓. Only x = 5/3 is valid, so exactly one solution exists.