Aimathic
Login | English | Deutsch

Free Math Worksheets

Build your own math worksheets from 21,000 problems for grades 3 to 12, from fractions to calculus. Every problem includes step-by-step solutions.

Absolute value in context

Click problems to add them to your worksheet.

5174996
A thermometer reads \(-12\,^\circ\text{F}\) one night and \(10\,^\circ\text{F}\) the next day. Write an inequality using absolute-value notation to show which temperature is farther from \(0\,^\circ\text{F}\).

Hints

- Absolute value gives a temperature’s distance from zero. - Compare the two distances. - Use \(>\) when the first distance is greater.

Solution

1. The distances from zero are \(|-12|=12\) and \(|10|=10\). 2. Since \(12>10\), the comparison is \(|-12|>|10|\).

Answer

\(|-12|>|10|\)
5175126
Five players have these scores after one round: Player A: \(-25\) points Player B: \(18\) points Player C: \(-12\) points Player D: \(22\) points Player E: \(-3\) points a) Which player’s score has the least absolute value? b) List the players from greatest to least absolute value of their scores.

Hints

- The absolute value of a score is its distance from zero. - Find all five absolute values before ordering the players. - Ignore the signs when comparing absolute values.

Solution

1. The absolute values of the scores are \(25,18,12,22,3\) for Players A through E. 2. The least absolute value is \(3\), so Player E answers part a). 3. From greatest to least absolute value, the order is Player A, Player D, Player B, Player C, Player E.

Answer

a) Player E b) Player A, Player D, Player B, Player C, Player E
5190976
Earth’s latitude lines form a scale from the South Pole to the North Pole. a) What is the angular distance from the North Pole to the South Pole? b) Research station \(A\) is at \(80^\circ\,\text{N}\), and station \(B\) is at \(10^\circ\,\text{N}\). Which station is closer to the North Pole? Justify your answer.

Hints

- Represent south latitudes with negative values and north latitudes with positive values. - Distance on a number line is the absolute value of the difference. - Compare each station’s distance from \(90^\circ\).

Solution

1. Represent the North Pole by \(90^\circ\) and the South Pole by \(-90^\circ\). Their distance is \(\lvert 90 - (-90) \rvert = 180^\circ\). 2. Station \(A\) is \(\lvert 80 - 90 \rvert = 10^\circ\) from the North Pole. 3. Station \(B\) is \(\lvert 10 - 90 \rvert = 80^\circ\) from the North Pole. 4. Since \(10^\circ < 80^\circ\), station \(A\) is closer.

Answer

a) \(180^\circ\) b) Station \(A\) is closer to the North Pole.
5191016
On a simplified map, Anchorage is shown at latitude \(61^\circ\,\text{N}\). The equator is at \(0^\circ\), and the North Pole is at \(90^\circ\,\text{N}\). Is Anchorage closer to the equator or to the North Pole? Justify your answer using the latitude values.

Hints

- Distance on a number line is the absolute value of the difference. - Find the distance from \(61^\circ\) to each reference latitude. - The smaller distance represents the closer location.

Solution

1. The angular distance from Anchorage to the equator is \(\lvert 61 - 0 \rvert = 61^\circ\). 2. The angular distance from Anchorage to the North Pole is \(\lvert 61 - 90 \rvert = 29^\circ\). 3. Since \(29^\circ < 61^\circ\), Anchorage is closer to the North Pole.

Answer

Anchorage is closer to the North Pole. It is \(29^\circ\) from the North Pole and \(61^\circ\) from the equator.
5191026
Two research stations are at different latitudes. Station Alpha is at \(20^\circ\,\text{N}\), and Station Beta is at \(75^\circ\,\text{N}\). The midpoint between the equator at \(0^\circ\) and the North Pole at \(90^\circ\,\text{N}\) is \(45^\circ\,\text{N}\). Which station is farther from this midpoint? Justify your answer by comparing the distances.

Hints

- Find each station’s distance from \(45^\circ\). - Distance is nonnegative, so compare absolute differences. - The greater distance identifies the station farther from the midpoint.

Solution

1. Station Alpha is \(\lvert 20 - 45 \rvert = 25^\circ\) from the midpoint. 2. Station Beta is \(\lvert 75 - 45 \rvert = 30^\circ\) from the midpoint. 3. Since \(30^\circ > 25^\circ\), Station Beta is farther from the midpoint.

Answer

Station Beta is farther from the midpoint. Station Alpha is \(25^\circ\) away, while Station Beta is \(30^\circ\) away.
5331446
Several points are marked on the coordinate plane. Write each coordinate pair. Which point or points are farthest from the x-axis?
Figure for problem 533144

Hints

- The vertical distance to the x-axis depends on the y-coordinate. - Distance is nonnegative, even for a point below the x-axis. - Compare the absolute values of the y-coordinates.

Solution

1. The coordinates are \(P(5, 1)\), \(Q(-4, -3)\), \(R(1, 4)\), \(S(-3, 1)\), and \(T(0, -4)\). 2. A point’s distance from the x-axis is the absolute value of its y-coordinate. 3. The distances are \(1, 3, 4, 1,\) and \(4\) units, respectively. 4. Points \(R\) and \(T\) are each \(4\) units from the x-axis, so they are tied for the greatest distance.

Answer

The coordinates are \(P(5, 1)\), \(Q(-4, -3)\), \(R(1, 4)\), \(S(-3, 1)\), and \(T(0, -4)\). Points \(R\) and \(T\) are farthest from the x-axis, at a distance of \(4\) units.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets