Question

appetizer (choose 3 of 4)
find the distance between the following numbers.

1. 5 and 4
2. - 3 and 7
3. 13 and - 4
4. - 1 and - 19

side dish (choose 2 of 3)
use the given number line to find the following lengths.

1. wx
2. wy
3. yx

main course (choose 3 of 4)
(for questions 1 and 2) if points a, b, and c are collinear with b between a and c, find the length indicated.

1. find ac if ab = 13 and bc = 9.
2. ac = 22, bc = x + 14, and ab = x + 10. find x.
3. find m∠wdc if m∠edc = 145° and m∠edw = 61°.
4. m∠vut = 175°, m∠vuj = 17x - 3, and m∠jut = 17x + 8. find x.

Explanation:

Appetizer

1. 5 and 4

Use the distance formula for two real - numbers \(d=\vert a - b\vert\). Here \(a = 5\) and \(b = 4\).
\(d=\vert5 - 4\vert=1\)

2. - 3 and 7

\(d=\vert-3 - 7\vert=\vert-10\vert = 10\)

3. 13 and - 4

\(d=\vert13-(-4)\vert=\vert13 + 4\vert=17\)

4. - 1 and - 19

\(d=\vert-1-(-19)\vert=\vert-1 + 19\vert=18\)

Side Dish

Assume \(W=-4\), \(X=-2\), \(Y = 4\) (from the number - line)

1. WX

\(WX=\vert-4-(-2)\vert=\vert-4 + 2\vert = 2\)

2. WY

\(WY=\vert-4 - 4\vert=\vert-8\vert = 8\)

3. YX

\(YX=\vert4-(-2)\vert=\vert4 + 2\vert=6\)

Main Course

1. If \(AB = 13\) and \(BC = 9\) and \(B\) is between \(A\) and \(C\), then by the segment - addition postulate \(AC=AB + BC\).

\(AC=13 + 9=22\)

2. Given \(AC = 22\), \(BC=x + 14\), and \(AB=x + 10\). Since \(AC=AB + BC\) (by the segment - addition postulate), then \((x + 10)+(x + 14)=22\).

Combine like terms: \(2x+24 = 22\).
Subtract 24 from both sides: \(2x=22 - 24=-2\).
Divide both sides by 2: \(x=-1\)

3. Given \(m\angle EDC = 145^{\circ}\) and \(m\angle EDW = 61^{\circ}\), then \(m\angle WDC=m\angle EDC-m\angle EDW\) (by the angle - addition postulate).

\(m\angle WDC=145^{\circ}-61^{\circ}=84^{\circ}\)

4. Given \(m\angle VUT = 175^{\circ}\), \(m\angle VUJ = 17x-3\), and \(m\angle JUT = 17x + 8\). Since \(m\angle VUT=m\angle VUJ+m\angle JUT\) (by the angle - addition postulate), then \((17x-3)+(17x + 8)=175\).

Combine like terms: \(34x+5 = 175\).
Subtract 5 from both sides: \(34x=175 - 5 = 170\).
Divide both sides by 34: \(x = 5\)

Answer:

Appetizer:

1. 1
2. 10
3. 17
4. 18

Side Dish:

1. 2
2. 8
3. 6

Main Course:

1. 22
2. \(x=-1\)
3. \(m\angle WDC = 84^{\circ}\)
4. \(x = 5\)