Free Math Worksheets

- Absolute value is a number's distance from zero. - An absolute value cannot be negative. - Distinguish a negative sign outside the absolute-value bars from one inside them.

1. For a), both absolute values equal \(\frac{7}{8}\), so the statement is true. 2. For b), \(\left|\frac{5}{6}\right| = \frac{5}{6}\), so the negative sign outside gives \(-\frac{5}{6}\). The statement is false. 3. For c), \(\left|-\frac{1}{2}\right| = \frac{1}{2} > 0\), so the statement is true. 4. For d), \(\left|-\frac{3}{4}\right| = \frac{3}{4}\), not \(-\frac{3}{4}\), so the statement is false.

a) True b) False c) True d) False

- Absolute value represents distance from zero. - A positive distance from zero usually corresponds to two opposite integers. - Remember that zero is an integer.

1. The integers \(-17\) and \(17\) are both \(17\) units from zero. 2. The integers less than \(3\) units from zero are \(-2,-1,0,1,2\). 3. Only \(0\) is zero units from zero.

a) \(-17\) and \(17\) b) \(-2,-1,0,1,2\) c) \(0\)

- Think of points the same distance from zero on opposite sides. - Consider a number and its opposite. - Zero is the only integer equal to its own opposite.

1. Absolute value represents distance from zero. 2. A nonzero integer and its opposite are different points that are the same distance from zero. 3. For example, \(|5|=5\) and \(|-5|=5\), so the answer is yes.

Yes. A nonzero integer and its opposite have the same absolute value. For example, \(|12|=|-12|=12\).

- Find each number’s distance from zero. - Compare the two nonnegative distances. - The sign of the original number does not make its distance negative.

1. \(|-12|=12\) and \(|10|=10\). 2. Since \(12>10\), \(-12\) is farther from zero.

\(-12\) is farther from zero because \(|-12|=12>|10|=10\).

- How do you find the distance between two points on a number line? - Sketch the pairs if needed. - Compare the four distances.

1. The distance in part a) is \(|1-(-12)|=13\). 2. The distance in part b) is \(|-1-(-13)|=12\). 3. The distance in part c) is \(|2-(-10)|=12\). 4. The distance in part d) is \(|7-(-5)|=12\). 5. The greatest distance is \(13\).

a) \(-12\) and \(1\)

- Two numbers can have the same positive distance from zero. - An absolute value cannot be negative. - Simplify the side without \(x\) first.

1. For a), absolute value is distance from zero. The two numbers at distance \(\frac{5}{12}\) from zero are \(\frac{5}{12}\) and \(-\frac{5}{12}\). 2. For b), absolute value is never negative, so there is no solution. 3. For c), first simplify the right side: \(\left\lvert-\frac{3}{7}\right\rvert = \frac{3}{7}\). Therefore, \(x = \frac{3}{7}\) or \(x = -\frac{3}{7}\).

a) \(x = \frac{5}{12}\) or \(x = -\frac{5}{12}\) b) No solution c) \(x = \frac{3}{7}\) or \(x = -\frac{3}{7}\)

- Absolute value represents distance from \(0\). - Two numbers can have the same nonzero distance from \(0\). - Find the distance from the negative point to the positive point.

1. Absolute value gives distance from \(0\), so the two numbers are \(5.2\) and \(-5.2\). 2. Their distance is \(5.2-(-5.2)=10.4\).

The numbers are \(-5.2\) and \(5.2\). The distance between them is \(10.4\).

- Think of distance from \(0\), not direction. - Include both positive and negative integers. - Do not forget \(0\).

1. Numbers less than \(4\) units from \(0\) have distances \(0\), \(1\), \(2\), or \(3\). 2. The integers are \(-3, -2, -1, 0, 1, 2, 3\). 3. There are \(7\) integers in the list.

The integers are \(-3, -2, -1, 0, 1, 2, 3\). There are \(7\) numbers.

- Compare the absolute values. - Convert fractions, decimals, and percents to a common form. - The smaller distance identifies the number closer to \(0\).

1. Distance from \(0\) is absolute value. 2. For a), \(\left|-\frac{5}{8}\right|=\frac{5}{8}=0.625\), while \(|0.65|=0.65\). Since \(0.625<0.65\), \(-\frac{5}{8}\) is closer to \(0\). 3. For b), \(|-1.2|=1.2\), and \(110\%=1.1\), so \(|110\%|=1.1\). Since \(1.1<1.2\), \(110\%\) is closer to \(0\).

a) \(-\frac{5}{8}\) b) \(110\%\)

- A midpoint is the same distance from both endpoints. - Find the distance from \(-5\) to each number in a pair. - Which pair gives equal distances?

1. For Pair A, the distance from \(-5\) to \(-12\) is \(7\), while the distance from \(-5\) to \(0\) is \(5\). The distances are not equal. 2. For Pair B, the distance from \(-5\) to \(-9\) is \(4\), and the distance from \(-5\) to \(-1\) is also \(4\). The distances are equal. 3. For Pair C, the distance from \(-5\) to \(-7\) is \(2\), while the distance from \(-5\) to \(-1\) is \(4\). The distances are not equal.

Pair B: \(-9\) and \(-1\)

- First find the number of unit intervals between the endpoints. - The interval crosses zero, so you can combine the distances from each endpoint to zero. - To find the midpoint, move half the total distance from one endpoint.

1. The distance between the numbers is \(6-(-14)=20\) units. At \(1\,\text{cm}\) per unit, the segment is \(20\,\text{cm}\) long. 2. Half of \(20\) is \(10\). Moving \(10\) units from \(-14\) gives \(-14+10=-4\).

a) \(20\,\text{cm}\) b) \(-4\)

- Picture the temperatures on a vertical number line. - Find the distance from \(-9\) to \(3\). - Move half the total distance from the colder temperature.

1. The distance from \(-9\) to \(3\) is \(3-(-9)=12\), so the temperature rose \(12\,^\circ\text{F}\). 2. Half of \(12\) is \(6\). Moving \(6\) degrees up from \(-9\) gives \(-9+6=-3\,^\circ\text{F}\).

a) \(12\,^\circ\text{F}\) b) \(-3\,^\circ\text{F}\)

- Absolute value is distance from zero. - For part a), consider values on both sides of zero. - For part b), keep only negative integers.

1. Distances greater than \(4\) and less than \(8\) are \(5,6,7\). The corresponding integers are \(-7,-6,-5,5,6,7\). 2. The negative integers less than \(5\) units from zero are \(-4,-3,-2,-1\).

a) \(-7,-6,-5,5,6,7\) b) \(-4,-3,-2,-1\)

- Opposites have the same distance from zero but different signs. - Absolute value tells a number’s distance from zero. - Consider which number lies exactly on the center of the number line.

1. The number whose opposite is \(-25\) is \(25\). 2. The negative number with absolute value \(14\) is \(-14\). 3. The positive number with absolute value \(8\) is \(8\). 4. The only number satisfying \(x=-x\) is \(0\).

a) \(25\) b) \(-14\) c) \(8\) d) \(0\)

- Interpret absolute value as distance from zero. - Each nonzero distance corresponds to two opposite integers. - The endpoint values \(-6\) and \(6\) are not included.

1. The condition \(|z|<6\) means the number is fewer than \(6\) units from zero. 2. The negative integers are \(-5,-4,-3,-2,-1\), followed by \(0\) and the positive integers \(1,2,3,4,5\). 3. Therefore, the ordered list is \(-5,-4,-3,-2,-1,0,1,2,3,4,5\).

\(-5,-4,-3,-2,-1,0,1,2,3,4,5\)

- First identify the possible absolute values. - Each positive absolute value corresponds to a positive and a negative integer. - Keep only the negative integers.

1. The whole-number distances strictly between \(3\) and \(8\) are \(4,5,6,7\). 2. Because the integers must be negative, the solutions are \(-7,-6,-5,-4\).

\(-7,-6,-5,-4\)

- Absolute value represents distance from zero. - Use \(=\) for equal distances. - Use \(<\) when the first number is closer to zero.

1. Distance from zero is represented by absolute value. 2. Equal distances give \(|-14|=|14|\). 3. Being closer to zero means having a lesser absolute value, so \(|-5|<|-9|\). 4. Similarly, \(|2|<|-3|\).

a) \(|-14|=|14|\) b) \(|-5|<|-9|\) c) \(|2|<|-3|\)

- Find the absolute value of each number first. - Order the resulting distances from least to greatest. - Replace each distance with its original absolute-value expression.

1. The distances from zero are \(|-22|=22\), \(|17|=17\), and \(|-11|=11\). 2. Since \(11<17<22\), the required chain is \(|-11|<|17|<|-22|\).

\(|-11|<|17|<|-22|\)

- For part a), compare the numbers’ positions on a number line. - For part b), find each number’s distance from zero. - Two opposite numbers have equal absolute values.

1. Ordering the integers themselves gives \(-12<-8<-5<0<5<7\). 2. Their absolute values are \(12,7,5,0,5,8\), respectively. 3. Ordering by absolute value gives \(0,-5,5,7,-8,-12\). The positions of \(-5\) and \(5\) may be reversed because their absolute values are equal.

a) \(-12<-8<-5<0<5<7\) b) \(0,-5,5,7,-8,-12\); \(-5\) and \(5\) may appear in either order.

- The lesser number lies farther left. - Absolute value gives distance from zero. - Position and distance answer different questions.

1. In part a), \(-22<18\), so \(-22\) lies farther left. Also, \(|-22|=22>|18|=18\), so \(-22\) has the greater distance from zero. 2. In part b), \(-50<-45\), so \(-50\) lies farther left. Also, \(|-50|=50>|-45|=45\), so \(-50\) has the greater distance from zero. 3. In part c), \(-5<15\), so \(-5\) lies farther left. However, \(|15|=15>|-5|=5\), so \(15\) has the greater distance from zero.

a) Farther left: \(-22\); greater distance: \(-22\) b) Farther left: \(-50\); greater distance: \(-50\) c) Farther left: \(-5\); greater distance: \(15\)

- Evaluate each absolute value before comparing. - The open side of a comparison symbol faces the greater value. - Pay attention to whether a number is inside absolute-value bars.

1. Since \(-13\) lies to the right of \(-15\), \(-13>-15\). 2. Both \(|-18|\) and \(|18|\) equal \(18\), so they are equal. 3. Since \(|-7|=7\) and \(7>6\), \(|-7|>6\). 4. Since \(|-1|=1\) and \(0<1\), \(0<|-1|\).

a) \(>\) b) \(=\) c) \(>\) d) \(<\)

- Find each number’s distance from zero. - Put the lesser distance first. - Use absolute-value bars to name the distances.

1. The distances from zero are \(|-8|=8\) and \(|-6|=6\). 2. Since \(6<8\), \(-6\) is closer to zero, and \(|-6|<|-8|\).

\(-6\) is closer to zero, and \(|-6|<|-8|\).

- First list the possible absolute values. - Include the upper endpoint but not the lower endpoint. - For each positive distance, include both opposite integers.

1. The possible integer absolute values are \(8,9,10,11\). 2. Each positive absolute value corresponds to two opposite integers. 3. Therefore, the integers are \(-11,-10,-9,-8,8,9,10,11\).

\(-11,-10,-9,-8,8,9,10,11\)

- Absolute value is distance from zero. - List the integer distances strictly between the two bounds. - Keep only natural numbers in part b).

1. The only integer fewer than \(1\) unit from zero is \(0\). 2. The whole-number absolute values strictly between \(99\) and \(103\) are \(100,101,102\). 3. These values are natural numbers, so they are the solutions to part b).

a) \(0\) b) \(100,101,102\)

- Absolute value is distance from zero. - Order the distances before returning to the original integers. - Write the original integers, not their absolute values, in part b).

1. The absolute values are \(34,12,5,0,42,17\). 2. Ordering those distances gives \(0<5<12<17<34<42\). 3. The corresponding original integers are \(0,-5,12,-17,-34,42\).

a) \(34,12,5,0,42,17\) b) \(0,-5,12,-17,-34,42\)

- List the possible whole-number distances first. - Each nonzero distance corresponds to a positive and a negative integer. - Neither endpoint is included.

1. The possible integer absolute values are \(13,14,15\). 2. Each positive absolute value corresponds to two opposite integers. 3. Therefore, the integers are \(-15,-14,-13,13,14,15\).

\(-15,-14,-13,13,14,15\)

- Think about how many unit intervals separate the two numbers. - Absolute value can represent distance. - Notice whether the numbers lie on the same side of zero or on opposite sides.

1. For part a), \(|-7-(-22)|=15\). 2. For part b), \(|18-(-12)|=30\). 3. For part c), \(|55-(-55)|=110\). 4. For part d), \(|-95-(-140)|=45\).

a) \(15\) b) \(30\) c) \(110\) d) \(45\)

- Picture the numbers on a number line. - Absolute value measures distance from zero. - Use the condition “to the left of zero” to choose between possible points.

1. Zero is its own opposite. 2. The two numbers \(-12\) and \(12\) are both \(12\) units from zero. 3. A point \(5\) units to the left of \(2\) is \(2-5=-3\). This is the solution because it lies left of zero. Its opposite is \(3\).

a) \(0\) b) \(-12\) and \(12\) c) The number is \(-3\), and its opposite is \(3\).

- Think of absolute value as a measured distance. - State what values an absolute value can have. - Compare that fact with the right side of the equation.

1. Absolute value represents distance from zero. 2. A distance is never negative, so \(|x|\ge0\) for every integer \(x\). 3. Therefore, \(|x|\) cannot equal \(-5\).

There is no integer solution because absolute value is always nonnegative, while \(-5\) is negative.

- Absolute value is distance from zero. - Find each absolute value before sorting. - Every absolute value is nonnegative.

1. The absolute values are \(24,13,35,7,2,0,19\). 2. From least to greatest, they are \(0<2<7<13<19<24<35\).

\(0,2,7,13,19,24,35\)

- Simplify the absolute-value expression first. - “Strictly between” excludes both endpoints. - List consecutive integers from left to right.

1. First evaluate the endpoint: \(|-3|=3\). 2. The integers satisfying \(-4<x<3\) are \(-3,-2,-1,0,1,2\).

\(-3,-2,-1,0,1,2\)

- Reverse the sign to find an opposite. - Compare the two points’ distances from \(0\). - Look for a pattern when adding each pair.

1. The opposites are \(5.4\), \(-3.1\), \(0\), and \(\frac{1}{2}\), respectively. 2. A number and its opposite are the same distance from \(0\) but lie on opposite sides, except that \(0\) is its own opposite. 3. Each sum is \(0\): \(-5.4+5.4=0\), \(3.1+(-3.1)=0\), \(0+0=0\), and \(-\frac{1}{2}+\frac{1}{2}=0\).

1. \(5.4\), \(-3.1\), \(0\), and \(\frac{1}{2}\) 2. They are the same distance from \(0\) and lie on opposite sides, unless the number is \(0\). 3. The sum is always \(0\).

- Do not include the endpoints. - Compare each integer’s absolute value with \(|2|\). - Look for distances greater than \(2\).

1. The integers strictly between the endpoints are \(-5,-4,-3,-2,-1,0,1\). 2. The distance of \(2\) from zero is \(2\). 3. Among the listed integers, \(-5,-4,-3\) have absolute values greater than \(2\).

The integers are \(-5,-4,-3,-2,-1,0,1\). Of these, \(-5,-4,-3\) are farther from zero than \(2\).

- Test each statement with a positive number, a negative number, and \(0\). - Recall how opposite numbers are placed on a number line. - Interpret absolute value as distance from \(0\).

1. Statement a) is false. If \(x=-5\), then \(-x=5\), which is positive. If \(x=0\), then \(-x=0\). 2. Statement b) is true. If \(x\ne0\), the numbers \(x\) and \(-x\) lie on opposite sides of \(0\) at the same distance. If \(x=0\), they are the same point. In every case, \(|x|=|-x|\). 3. Statement c) is false. Absolute value is distance from \(0\), and \(|0|=0\).

a) False. For example, if \(x=-5\), then \(-x=5\). b) True. Opposite numbers have equal distance from \(0\). c) False. \(|0|=0\).

- Absolute value is never negative. - The number with the smaller absolute value is closer to \(0\). - Find the value halfway between the two distances. - A positive absolute value corresponds to two opposite numbers.

1. \(|a|=|-7|=7\) and \(|b|=|3|=3\). 2. Since \(3<7\), the number \(b=3\) is closer to \(0\). 3. The value halfway between \(3\) and \(7\) is \(\frac{3+7}{2}=5\). The numbers with absolute value \(5\) are \(-5\) and \(5\).

a) \(|a|=7\) and \(|b|=3\) b) \(b=3\) c) \(-5\) and \(5\)

- First identify the closest integer to each number. - Distance is always nonnegative. - For a fraction, write the nearby integer with the same denominator before subtracting.

1. For a), the nearest integer is \(4\). The distance is \(|4-3.8|=0.2\). 2. For b), \(-1\frac{1}{4}=-1.25\), and the nearest integer is \(-1\). The distance is \(|-1-(-1.25)|=0.25\). 3. For c), \(-\frac{22}{7}\approx-3.14\), and the nearest integer is \(-3=-\frac{21}{7}\). The distance is \(\left|-\frac{21}{7}-\left(-\frac{22}{7}\right)\right|=\frac{1}{7}\). 4. For d), the nearest integer is \(0\). The distance is \(|0.45-0|=0.45\).

a) \(0.2\) b) \(0.25\) c) \(\frac{1}{7}\) d) \(0.45\)

- First find how many integer intervals lie between the two values. - Each interval represents \(2\,\text{cm}\), not \(1\,\text{cm}\). - You can use zero as an intermediate point when the values have different signs.

1. In part a), the numbers are \(13\) units apart because \(|8-(-5)|=13\). The segment length is \(13\times2\,\text{cm}=26\,\text{cm}\). 2. In part b), the numbers are \(9\) units apart because \(|-3-(-12)|=9\). The segment length is \(9\times2\,\text{cm}=18\,\text{cm}\).

a) \(26\,\text{cm}\) b) \(18\,\text{cm}\)

- A fixed distance can extend in either direction from a point. - Which operation moves to the right, and which operation moves to the left?

1. Move \(25\) units to the right: \(-14+25=11\). 2. Move \(25\) units to the left: \(-14-25=-39\).

The two possible coordinates are \(11\) and \(-39\).

- Find the difference between the coordinates for each distance. - A distance is always nonnegative. - The distance between the outer points can also be found by combining the two distances from the middle point.

1. The distance from \(R\) to \(S\) is \(|4-(-18)|=22\) units. 2. The distance from \(S\) to \(T\) is \(|22-4|=18\) units. 3. Since \(22>18\), point \(R\) is farther from \(S\). 4. The distance from \(R\) to \(T\) is \(|22-(-18)|=40\) units.

a) Point \(R\) is farther from \(S\): \(22\) units compared with \(18\) units. b) \(40\) units

- Picture each pair on a number line. - Find the total distance between the endpoints. - Move half that distance from the lesser endpoint. - The midpoint must be the same distance from both endpoints.

1. For part a), the distance is \(12\) units. Half of \(12\) is \(6\), and \(-14+6=-8\). 2. For part b), the distance is \(12\) units. Half of \(12\) is \(6\), and \(-3+6=3\). 3. For part c), the endpoints are opposites, so their midpoint is \(0\). 4. For part d), the distance is \(30\) units. Half of \(30\) is \(15\), and \(-10+15=5\).

a) \(-8\) b) \(3\) c) \(0\) d) \(5\)

- If \(M\) is the midpoint, how are the distances \(AM\) and \(MB\) related? - Find the distance from \(A\) to \(M\). - Continue the same distance and direction from \(M\) to locate \(B\).

1. The distance from \(A\) to \(M\) is \(2-(-6)=8\) units. 2. Because \(M\) is the midpoint, \(B\) must be \(8\) units on the other side of \(M\). 3. Move \(8\) units right from \(2\): \(2+8=10\).

Point \(B\) is at \(10\).

- Translate each absolute-value condition into distances from zero. - “At most” includes the stated endpoint. - For part c), count zero once and each nonzero absolute value as an opposite pair.

1. The only whole-number absolute values strictly between \(9\) and \(12\) are \(10\) and \(11\). Therefore, \(z=-11,-10,10,11\). 2. The integers with \(|z|\le2\) are \(-2,-1,0,1,2\). 3. The integers with \(|z|<6\) are zero and the five opposite pairs with absolute values \(1\) through \(5\). Thus, there are \(1+5\times2=11\) integers.

a) \(-11,-10,10,11\) b) \(-2,-1,0,1,2\) c) \(11\)

- Organize each case by number, absolute value, and opposite. - Absolute value is distance from zero. - Opposites are the same distance from zero on different sides.

1. For \(-31\), the absolute value is \(31\), and the opposite is \(31\). 2. The negative number with absolute value \(19\) is \(-19\). Its opposite is \(19\). 3. If the opposite is \(50\), the original number is \(-50\). Its absolute value is \(50\). 4. Seven units to the right of zero is \(7\). Its absolute value is \(7\), and its opposite is \(-7\).

a) Number: \(-31\); absolute value: \(31\); opposite: \(31\) b) Number: \(-19\); absolute value: \(19\); opposite: \(19\) c) Number: \(-50\); absolute value: \(50\); opposite: \(50\) d) Number: \(7\); absolute value: \(7\); opposite: \(-7\)

- For part a), compare each number’s distance from zero with \(15\). - For part b), compare the original signed values, not their absolute values. - Among negative numbers, a number with greater absolute value is less.

1. The absolute values are \(22,14,5,30,35,9\). 2. The original numbers with absolute value greater than \(15\) are \(-22,30,-35\). 3. Ordering those numbers themselves from greatest to least gives \(30>-22>-35\).

a) \(-22,30,-35\) b) \(30>-22>-35\)

- Order the signed values separately from their absolute values. - Absolute value measures distance from zero. - Zero has the least possible absolute value.

1. Ordering the signed values gives \(-17<-14<-2<0<9<13\). 2. Their absolute values are \(14,9,2,0,13,17\). 3. The greatest absolute value is \(17\), which belongs to \(-17\). The least absolute value is \(0\), which belongs to \(0\).

a) \(-17<-14<-2<0<9<13\) b) \(-17\) c) \(0\)

- “Greatest to least” means descending order. - Check each number’s distance from zero for part b). - A value with absolute value exactly \(20\) does not qualify.

1. Ordering the signed values from greatest to least gives \(20>15>8>-2>-15>-38>-40\). 2. The absolute values are \(40,15,15,8,2,38,20\). 3. The numbers with absolute value strictly less than \(20\) are \(15,-15,8,-2\).

a) \(20>15>8>-2>-15>-38>-40\) b) \(15,-15,8,-2\)

- Opposite numbers have equal absolute values. - Write each absolute value before ordering. - Tied absolute values can appear in either order.

1. The absolute values are \(9,14,21,0,14,5,2\). 2. The numbers \(14\) and \(-14\) have the same absolute value. 3. Ordering the original numbers by absolute value gives \(0,-2,5,-9,14,-14,-21\). The positions of \(14\) and \(-14\) may be reversed.

a) \(14\) and \(-14\) b) \(0,-2,5,-9,14,-14,-21\); \(14\) and \(-14\) may appear in either order.

- First identify integers whose distance from zero exceeds \(4997\). - “Strictly between” excludes both endpoints. - Check the candidates against both conditions.

1. The integers with absolute value greater than \(4997\) near the interval endpoints include \(-5000,-4999,-4998\) and \(4998,4999,5000\). 2. Because the integers must lie strictly between \(-5000\) and \(5000\), both endpoints are excluded. 3. The remaining integers are \(-4999,-4998,4998,4999\).

\(-4999,-4998,4998,4999\)

- Test a positive integer, a negative integer, and zero. - Interpret absolute value as distance from zero. - Compare the cases \(x\ge0\) and \(x<0\).

1. If \(x\ge0\), its distance from zero equals its value, so \(|x|=x\). 2. If \(x<0\), then \(|x|\) is positive while \(x\) is negative, so \(|x|>x\). 3. Thus, \(|x|\) is either equal to or greater than \(x\), and it can never be less than \(x\).

a) All nonnegative integers, so \(x\ge0\) b) For \(x\ge0\), \(|x|=x\). For \(x<0\), \(|x|>x\). Therefore, \(|x|<x\) has no integer solutions.

- Find every number’s distance from zero. - The least distance identifies part a). - Keep absolute-value notation in the inequality chain.

1. The absolute values are \(12,5,8,2,1,10\). 2. The least absolute value is \(1\), which belongs to \(-1\). The greatest is \(12\), which belongs to \(-12\). 3. The ordered expressions are \(|-1|<|2|<|5|<|-8|<|10|<|-12|\).

a) \(-1\) b) \(-12\) c) \(|-1|<|2|<|5|<|-8|<|10|<|-12|\)

- Use the distance between coordinates for parts a) and b). - Absolute value gives a number's distance from zero. - Compare the three absolute values.

1. The distance between \(a\) and \(b\) is \(|12-(-32)|=44\). 2. The distance between \(b\) and \(c\) is \(|-58-12|=70\). 3. The distances from zero are \(|a|=32\), \(|b|=12\), and \(|c|=58\). Since \(58\) is greatest, \(c\) is farthest from zero.

a) \(44\) b) \(70\) c) \(c=-58\), because \(|-58|=58\) is the greatest absolute value.

- Interpret absolute value as distance from zero. - Test each statement with zero as well as positive and negative integers. - A single counterexample disproves an “always” statement.

1. Statement a) is true. Absolute value is distance from zero, so the absolute value of an integer is a nonnegative integer. 2. Statement b) is true because \(a+(-a)=0\) for every integer \(a\). 3. Statement c) is false. For positive numbers and zero, the absolute value equals the number, such as \(|5|=5\).

a) True; an integer’s absolute value is a nonnegative integer. b) True; \(a+(-a)=0\). c) False; for example, \(|5|=5\), not a greater value.

- A number’s distance from zero is its absolute value. - Among negative numbers, the one with the greater absolute value is less. - Compare each number’s distance from \(-5\).

1. The distances from zero are the absolute values. Since \(|-14|=14\) and \(|11|=11\), the numbers more than \(10\) units from zero are \(-14\) and \(11\). 2. From least to greatest, the numbers are \(-14<-6<-3<2<9<11\). 3. The distance from \(-6\) to \(-5\) is \(1\), which is less than the distance from any other listed number to \(-5\). Therefore, \(-6\) is closest.

a) \(-14\) and \(11\) b) \(-14<-6<-3<2<9<11\) c) \(-6\)

- Split the total distance equally on both sides of \(0\). - Opposite numbers have \(0\) as their midpoint. - Apply the opposite operation one step at a time.

1. Opposite numbers are the same distance from \(0\). Each is \(15\div2=7.5\) units from \(0\). The ordered pairs are \((7.5, -7.5)\) and \((-7.5, 7.5)\). 2. The midpoint of opposite numbers is always \(0\), so \(r=0\). 3. The opposite of \(-8.3\) is \(8.3\), and the opposite of \(8.3\) is \(-8.3\).

1. \((7.5, -7.5)\) and \((-7.5, 7.5)\) 2. \(r=0\) 3. \(-8.3\)

- Picture equal distances extending in both directions from zero. - For \(|x|>12\), look beyond both \(-12\) and \(12\). - Use the definition of absolute value for the explanation.

1. The integers fewer than \(5\) units from zero are \(-4,-3,-2,-1,0,1,2,3,4\). 2. Being more than \(12\) units from zero means \(x\le-13\) or \(x\ge13\). 3. The statement is true because absolute value is defined as distance from zero.

1. \(-4,-3,-2,-1,0,1,2,3,4\) 2. \(x\le-13\) or \(x\ge13\) 3. True; absolute value measures distance from zero.

- Find the total distance between the endpoints. - Move half that distance from one endpoint to find the midpoint. - When counting integers between the endpoints, do not include the endpoints.

1. The total distance is \(4-(-8)=12\) units. 2. Half of \(12\) is \(6\), and \(-8+6=-2\), so the midpoint is \(-2\). 3. The integers strictly between the endpoints are \(-7,-6,-5,-4,-3,-2,-1,0,1,2,3\), for a total of \(11\).

The midpoint is \(-2\), and \(11\) integers lie strictly between the endpoints.

- Determine the value represented by one small tick interval. - Values left of zero are negative, and values right of zero are positive. - Count from a nearby labeled tick to each marker. - Absolute value is the distance from zero and is never negative.

1. There are \(5\) equal intervals between consecutive labeled multiples of \(10\), so each small interval represents \(2\). 2. Point \(X\) is at \(-24\), point \(Y\) is at \(-8\), and point \(Z\) is at \(16\). 3. Their distances from zero are \(|-24|=24\), \(|-8|=8\), and \(|16|=16\).

1. \(X=-24\), \(Y=-8\), and \(Z=16\) 2. \(|X|=24\), \(|Y|=8\), and \(|Z|=16\)

- Read the coordinate of each marked point. - Use distance on a number line to find each segment length. - Pay attention to signs when subtracting coordinates. - Compare the two lengths after calculating them.

1. Read the coordinates: \(P=-12\), \(Q=-4\), and \(R=9\). 2. The length of \(PQ\) is \(|-4-(-12)|=8\,\text{cm}\). 3. The length of \(QR\) is \(|9-(-4)|=13\,\text{cm}\). 4. Since \(13-8=5\), segment \(QR\) is \(5\,\text{cm}\) longer.

Segment \(QR\) is \(5\,\text{cm}\) longer than segment \(PQ\).

- Values left of zero are negative. - A number's distance from zero is its absolute value.

1. Each small tick interval represents \(1\). The marked coordinates are \(A=-17\), \(B=-8\), and \(C=4\). 2. Their distances from zero are \(|-17|=17\), \(|-8|=8\), and \(|4|=4\). 3. Since \(17\) is greatest, \(A=-17\) is farthest from zero.

a) \(A=-17\), \(B=-8\), \(C=4\) b) \(A=-17\) is farthest from zero.

- First convert the length in centimeters to number-line units. - How far is each endpoint from the midpoint? - Move the same distance left and right from \(-5\).

1. Convert the physical length to number-line units: \(6\times2=12\) units. 2. Each endpoint is half the total distance from the midpoint: \(12\div2=6\) units. 3. The left endpoint is \(-5-6=-11\), and the right endpoint is \(-5+6=1\).

The endpoints are \(-11\) and \(1\).

- A nonzero distance from zero corresponds to two opposite numbers. - Think about what taking an opposite does to a sign. - A distance cannot be negative.

1. The two numbers \(-21\) and \(21\) are both \(21\) units from zero. 2. If the opposite of \(x\) is \(-4\), then \(x=4\), and \(|x|=4\). 3. The statement is true because absolute value represents distance, and distance cannot be negative. 4. Since \(|-3|=3\), the number must have absolute value \(0\), \(1\), or \(2\). It cannot be \(-1\) or \(-2\), so \(0\), \(1\), or \(2\) works.

a) \(-21\) and \(21\) b) \(x=4\) and \(|x|=4\) c) True; absolute value is a distance and is therefore nonnegative. d) One possible answer is \(0\). Other possible answers are \(1\) and \(2\).

- A counterexample satisfies the condition but makes the conclusion false. - Consider choosing a negative value for \(a\). - A number can have a large absolute value and still be less than a positive number.

1. Choose \(a=-10\) and \(b=5\). 2. Then \(|a|=10\) and \(|b|=5\), so \(|a|=2|b|\). 3. However, \(-10<5\), so \(a<b\). This counterexample disproves the claim.

One counterexample is \(a=-10\) and \(b=5\). Then \(|a|=10=2\times5=2|b|\), but \(a<b\) because \(-10<5\).

- Find the opposite and absolute value of each example separately. - Compare what happens to positive and negative signs. - Remember that \(0\) is its own opposite.

1. For \(+7\), removing the plus sign leaves \(7\), but the opposite is \(-7\). Leon’s rule fails. 2. For \(-12\), removing the negative sign gives \(12\), which happens to be both the opposite and the absolute value of \(-12\). 3. The opposite of \(0\) is \(0\), and \(|0|=0\). 4. To find an opposite, change the sign, except that \(0\) remains \(0\). Absolute value is distance from zero and is never negative.

Sophie is correct. The opposite is found by changing a number’s sign, with \(0\) as its own opposite. Absolute value is the number’s distance from zero. Removing a negative sign gives the absolute value only when the number is written as negative.

- For the counterexample, try one negative and one positive integer. - Among negative integers, the lesser value lies farther left. - Equal absolute values mean equal distances from zero.

1. For example, let \(a=-10\) and \(b=2\). Then \(-10<2\), but \(|-10|=10>|2|=2\), so the claim is false. 2. Because \(a<b<0\), \(a\) lies farther left and therefore farther from zero. Thus, \(|a|>|b|\). 3. Two different numbers with equal absolute value lie on opposite sides of zero at equal distances. They are opposites.

a) One counterexample is \(a=-10\), \(b=2\): \(a<b\), but \(|a|>|b|\). b) \(a\) has the greater absolute value because it lies farther from zero. c) The points are symmetric about zero.