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Absolute Value: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 A small expression

Evaluate $\lvert -(-16) \rvert$ .
1. Hint 1
  The bars report how far the enclosed number lies from zero.
2. Hint 2
  The minus sign before the parentheses asks for the opposite of the number in parentheses; settle that instruction before measuring.
Answer

$16$ .

Full solution

The opposite of a negative number lies the same distance to the right of zero.
$-(-16) = 16$

That point is sixteen units from zero, so the bars give its distance.
$\lvert 16 \rvert = 16$

The result is zero or positive, as a distance must be.

Answer

$16$ .

Key idea

Complete an instruction inside the bars before measuring the resulting number's distance from zero.
Problem 1: Mark complete
Problem 2 A product in bars

Evaluate $\lvert 0 \times (-19) \rvert$ .
1. Hint 1
  The bars act after the expression inside has become one number.
2. Hint 2
  Read the multiplication as a count of equal groups, then consider where its total sits on the number line.
Answer

$0$ .

Full solution

There are zero groups of the negative number, so the product is zero.
$0 \times (-19) = 0$

Zero is at the reference point itself, with no distance to count.
$\lvert 0 \rvert = 0$

Answer

$0$ .

Key idea

An absolute value can equal zero, since zero has no distance from itself.
Problem 2: Mark complete
Problem 3 Either side of the gate

A straight running track has integer position labels, with one meter between neighboring labels, and a gate at the label $0$ . Ana stands at $-11$ . Ben stands on the other side of the gate, exactly as far from it as Ana. What is Ben's label, and how many meters apart are Ana and Ben?
1. Hint 1
  The side of the gate and the distance from it are different pieces of information.
2. Hint 2
  Find Ana's distance from the gate, then place Ben that far away on the other side.
3. Hint 3
  The gap between two labels is the absolute value of their difference.
Answer

Ben is at $11$ ; they are $22$ meters apart.

Full solution

Ana's distance from the gate is the absolute value of her label.
$\lvert -11 \rvert = 11$

Ben is also $11$ meters from the gate but on the positive side, so his label is the opposite of Ana's, $11$ .

The gap is the absolute value of the difference of the labels.
$\begin{aligned} 11 - (-11) &= 22 \\ \lvert 22 \rvert &= 22 \end{aligned}$

They are $22$ meters apart.
This checks: eleven meters from Ana to the gate and eleven more to Ben.

Answer

Ben is at $11$ ; they are $22$ meters apart.

Key idea

A number and its opposite sit the same distance from zero on opposite sides.
Problem 3: Mark complete
Problem 4 A quotient, then a sum

Evaluate $\lvert (-21) \div 3 + 2 \rvert$ .
1. Hint 1
  Everything between one pair of bars must become a single number before its distance is measured.
2. Hint 2
  Within the bars, division comes before addition.
3. Hint 3
  Find the signed quotient, add the remaining term, and then count the resulting number's distance from zero.
Answer

$5$ .

Full solution

Divide first inside the bars.
The signs differ, so the quotient is negative.
$(-21) \div 3 = -7$

Adding two moves two units right from that quotient.
$-7 + 2 = -5$

Now measure the resulting number rather than the original terms separately.
$\lvert -5 \rvert = 5$

The division checks by multiplication: $(-7) \times 3 = -21$ , and the final point is five units from zero.

Answer

$5$ .

Key idea

The arithmetic inside the bars follows the usual order of operations before the distance is measured.
Problem 4: Mark complete
Problem 5 Two pairs of bars

Evaluate $-\lvert 8 - (-6) \rvert + \lvert (-2) \times 3 \rvert$ .
1. Hint 1
  Each pair of bars encloses its own calculation, and symbols outside the pair keep their separate jobs.
2. Hint 2
  Work out the subtraction inside the first pair and the multiplication inside the second pair.
3. Hint 3
  Measure each result, apply the minus sign in front of the first pair, and then add the two signed values.
Answer

$-8$ .

Full solution

Subtracting a negative adds its opposite, so the first enclosed calculation is $8+6$ .
$8 - (-6) = 14$

The first bars give $14$ , and the minus sign outside them then takes its opposite.
$-\lvert 14 \rvert = -14$

The second enclosed calculation has factors with different signs.
$(-2) \times 3 = -6$

Measure that product, then combine the two contributions.
$\begin{aligned} \lvert -6 \rvert &= 6 \\ -14 + 6 &= -8 \end{aligned}$

The final sum checks by undoing the addition:
$-8 - 6 = -14.$

Answer

$-8$ .

Key idea

A minus sign outside a pair of bars takes the opposite of the measured distance.
Problem 5: Mark complete
Problem 6 The inspection elevator

An elevator moves vertically in a shaft. Heights are measured in meters from the entrance, with negative heights below it. The elevator travels from $-12$ to $-3$ , then from $-3$ to $-8$ , without any other stops or reversals. What total distance does it travel?
1. Hint 1
  Total travel counts the length of each part of the trip, even when the direction changes.
2. Hint 2
  Find how far apart the starting and finishing heights of each part are.
3. Hint 3
  Write each length as the absolute value of a difference, then add the lengths.
Answer

$14$ meters.

Full solution

For the first part, subtract the starting height from the finishing height and measure the result.
$\begin{aligned} -3 - (-12) &= 9 \\ \lvert 9 \rvert &= 9 \end{aligned}$

The first part is $9$ meters.
Repeat for the second part, which travels downward.
$\begin{aligned} -8 - (-3) &= -5 \\ \lvert -5 \rvert &= 5 \end{aligned}$

The second part is $5$ meters, so the total length is their sum.
$9 + 5 = 14$

The elevator travels $14$ meters in all.
Its final height checks:
$-12+9-5=-8.$

Answer

$14$ meters.

Key idea

Add the distances of separate parts of a journey to find total travel, even when their signed changes have different signs.
Problem 6: Mark complete
Problem 7 Along the cable

A straight cable has position labels measured in meters from a reference mark. A sensor is at $-4$ , beacon A is at $-15$ , and beacon B is at $9$ . Which beacon is closer to the sensor, and by how many meters?
1. Hint 1
  Closeness here concerns the gap from the sensor, not the gap from the reference mark.
2. Hint 2
  Subtract the sensor position from each beacon position, then measure each resulting difference.
3. Hint 3
  Compare the two lengths and find the difference between the longer and shorter lengths.
Answer

Beacon A is closer by $2$ meters.

Full solution

Find the gap from the sensor to beacon A.
$\begin{aligned} -15 - (-4) &= -11 \\ \lvert -11 \rvert &= 11 \end{aligned}$

Beacon A is $11$ meters away.
Find the gap to beacon B in the same way.
$\begin{aligned} 9 - (-4) &= 13 \\ \lvert 13 \rvert &= 13 \end{aligned}$

Beacon B is $13$ meters away, so beacon A is closer.
The difference in the distances is
$13 - 11 = 2$

Beacon A is closer by $2$ meters.
Counting left eleven meters from the sensor reaches beacon A; counting right thirteen meters reaches beacon B.

Answer

Beacon A is closer by $2$ meters.

Key idea

To compare distances from a point that is not zero, measure differences from that point.
Problem 7: Mark complete
Problem 8 The machine display

A machine accepts an integer $x$ and displays $\lvert x \rvert$ . A technician changes its input from $x$ to $-x$ . Can this change the displayed number? Explain your decision for every integer, including zero.
1. Hint 1
  Ask which information about a number's position is kept by the bars.
2. Hint 2
  Compare the locations of $x$ and $-x$ relative to zero, then consider what happens when the input is zero itself.
Answer

No; $\lvert x \rvert = \lvert -x \rvert$ for every integer $x$ , including $0$ .

Full solution

For a nonzero integer, changing $x$ to $-x$ takes its opposite.
The point moves to the other side of zero but remains the same distance from zero.
Each displayed value records that distance, so the values agree.

When the input is zero, taking its opposite leaves it at zero.
$\begin{aligned} -0 &= 0 \\ \lvert 0 \rvert &= 0 \end{aligned}$

Thus there is no integer input for which the change alters the display.
$\lvert x \rvert = \lvert -x \rvert$

Answer

No; $\lvert x \rvert = \lvert -x \rvert$ for every integer $x$ , including $0$ .

Key idea

A number and its opposite have the same absolute value since they share a distance from zero.
Problem 8: Mark complete
Problem 9 Two proposed replacements

For an integer $x$ , Kai replaces $\lvert x \rvert$ with $x$ , and Leena replaces it with $-x$ . State exactly which inputs make each replacement correct. Could either replacement be used for every integer? Also determine the sign of Leena's output when $x<0$ , and explain how you know.
1. Hint 1
  Check whether each proposed output represents the distance of the input from zero.
2. Hint 2
  The minus sign in $-x$ asks for an opposite; it does not by itself tell you whether the resulting number is positive or negative.
3. Hint 3
  Test a positive input, a negative input and zero, then describe the inputs that work for each replacement.
Answer

Kai: $x \ge 0$ (zero or positive). Leena: $x<0$ or $x=0$ (zero or negative; $x \le 0$ is the same). Neither replacement works for every integer. For $x<0$ , $-x>0$ .

Full solution

For a positive input, the distance is the number itself, so Kai's replacement works.
Leena's replacement gives a negative output, which is not a distance.
For example, at $x=14$ :
$\begin{aligned} \lvert 14 \rvert &= 14 \\ -(14) &= -14 \end{aligned}$

For a negative input, Kai's output is negative, so it is not the distance.
Leena takes the opposite of that negative input, moving it to the positive side while keeping the same distance.
At $x=-14$ :
$\begin{aligned} -(-14) &= 14 \\ \lvert -14 \rvert &= 14 \end{aligned}$

Both replacements also work for zero, since $-0=0$ and
$\lvert0\rvert=0.$
Thus Kai is correct exactly when $x\ge0$ , and Leena is correct exactly when $x<0$ or $x=0$ .

The positive and negative inputs need different choices: return $x$ for $x\ge0$ , and return $-x$ for $x<0$ .
In the negative case, $-x$ is positive because it is the opposite of a negative number.
So neither single replacement works for every integer.

Answer

Kai: $x \ge 0$ (zero or positive). Leena: $x<0$ or $x=0$ (zero or negative; $x \le 0$ is the same). Neither replacement works for every integer. For $x<0$ , $-x>0$ .

Key idea

The two cases keep zero and positive inputs unchanged and replace negative inputs with their positive opposites.
Problem 9: Mark complete
Problem 10 Tess checks her work

Tess evaluates $\lvert 5 + (-4) \times 3 \rvert$ by replacing $-4$ with $4$ , multiplying, and then adding. She reports $17$ . Is her calculation valid? Explain and give the correct value of the expression.
1. Hint 1
  A pair of bars measures the complete expression it encloses.
2. Hint 2
  Keep the signs of the terms while doing the multiplication inside the bars.
3. Hint 3
  After adding the signed product to the remaining term, measure the distance of that single result from zero.
Answer

No; the correct value is $7$ .

Full solution

Tess changes a number inside the bars before completing the enclosed calculation.
The bars measure the result of the entire sum, not the separate negative factor.

First multiply with the original sign intact.
$(-4) \times 3 = -12$

Now add inside the bars.
$5 + (-12) = -7$

Finally measure the resulting number's distance from zero.
$\lvert -7 \rvert = 7$

The calculation is not valid, and the correct value is $7$ .
The internal sum lies seven units to the left of zero, which checks the final distance.

Answer

No; the correct value is $7$ .

Key idea

The bars measure the result of the enclosed calculation, so signs inside them must stay intact until that calculation is complete.
Problem 10: Mark complete

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