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Absolute value applications

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5122117
Each statement about rational numbers is false. Give a counterexample for each statement. a) The absolute value of a sum is always equal to the sum of the absolute values. b) Adding a rational number to its absolute value always gives \(0\). c) If \(a < b\), then \(|a| < |b|\).

Hints

- For a), try one positive number and one negative number. - For b), test a positive number. - For c), choose a negative number far from zero and a small positive number. - One valid counterexample is enough to disprove a universal statement.

Solution

1. For a), let \(a = 2\) and \(b = -5\). Then \(|2 + (-5)| = 3\), but \(|2| + |-5| = 7\). Therefore, the statement is false. 2. For b), let \(a = 4\). Then \(4 + |4| = 8 \ne 0\), so the statement is false. 3. For c), let \(a = -10\) and \(b = 2\). Although \(-10 < 2\), \(|-10| = 10 > 2 = |2|\). Therefore, the statement is false.

Answer

a) For example, \(|2 + (-5)| = 3\), but \(|2| + |-5| = 7\). b) For example, \(4 + |4| = 8\), not \(0\). c) For example, \(-10 < 2\), but \(|-10| > |2|\).
5185477
Evaluate each expression step by step. a) \(|-42|+58-(-100)\) b) \(|15-40|+(12-30)\)

Hints

- Evaluate inside absolute value bars and parentheses first. - Absolute value is a number’s distance from zero, so it is never negative. - Subtracting a negative number is the same as adding a positive number.

Solution

1. In a), \(|-42|=42\), so \(42+58-(-100)=42+58+100=200\). 2. In b), \(15-40=-25\), so \(|15-40|=25\). Also, \(12-30=-18\). Therefore, \(25+(-18)=7\).

Answer

a) \(200\) b) \(7\)
5185487
Evaluate each expression. a) \(|150-210|-(45-90)\) b) \(|-25-15|+[-30-(-50)]\)

Hints

- Treat absolute value bars like grouping symbols and evaluate inside them first. - Keep careful track of signs when subtracting a negative number. - Write intermediate values before combining them.

Solution

1. In a), \(150-210=-60\), so \(|150-210|=60\). Also, \(45-90=-45\). Therefore, \(60-(-45)=105\). 2. In b), \(-25-15=-40\), so \(|-25-15|=40\). Also, \(-30-(-50)=20\). Therefore, \(40+20=60\).

Answer

a) \(105\) b) \(60\)
5185787
Which tasks have a value of \(30\)? (A) \(|-18|+|-12|\) (B) \(18+12\) (C) Add the absolute values of \(-18\) and \(-12\). (D) Subtract \(-12\) from \(18\). (E) \(18-(-12)\) (F) Find the distance between \(18\) and \(-12\) on a number line. (G) \(-30+60\) (H) \(-18-12\)

Hints

- Absolute value gives distance from zero. - The distance between two numbers is the absolute value of their difference. - Subtracting a negative number is the same as adding a positive number.

Solution

1. A, B, and C each equal \(18+12=30\). 2. D and E each equal \(18-(-12)=30\). 3. The distance in F is \(|18-(-12)|=30\). 4. G equals \(-30+60=30\), while H equals \(-30\).

Answer

A, B, C, D, E, F, and G
5185497
Evaluate each expression. a) \(|120-450+130|-[-100-(250-600)]\) b) \(|-55|+(-125)-|200-350|-(75-200)\)

Hints

- Work from the innermost grouping symbols outward. - Record the value of each absolute value and parenthetical expression. - Check every sign before combining the intermediate results.

Solution

1. In a), \(120-450+130=-200\), so the absolute value is \(200\). Also, \(250-600=-350\), and \(-100-(-350)=250\). Therefore, \(200-250=-50\). 2. In b), \(|-55|=55\), \(|200-350|=150\), and \(75-200=-125\). Therefore, \(55-125-150-(-125)=-95\).

Answer

a) \(-50\) b) \(-95\)

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