Question

Write without the absolute value sign: |z−6|−|z−5|, if z<5.

Answer 1

The expression |z-6|-|z-5|, when z<5, simplifies to 1 by considering the definition of absolute value and the given condition that z is less than 5.

Step-by-step explanation:

To rewrite the expression without the absolute value signs for z<5, we need to consider the definition of absolute value and the given condition. The absolute value of a number a, denoted as |a|, is the distance of a from zero on the number line, regardless of direction. Therefore, if a is positive, |a|=a, and if a is negative, |a|=-a.

Given that z<5, both z-6 and z-5 are negative because z is less than both 5 and 6. Hence, we rewrite |z-6| as -(z-6) and |z-5| as -(z-5).

The original expression |z-6|-|z-5| becomes:

- (z - 6) - (-(z - 5)) = -z + 6 - (-z + 5) = -z + 6 + z - 5

The z terms cancel out, simplifying the expression to: 1

Answer 2

The expression |z-6|-|z-5|, when z<5, simplifies to 1 by considering the definition of absolute value and the given condition that z is less than 5.

Step-by-step explanation:

To rewrite the expression without the absolute value signs for z<5, we need to consider the definition of absolute value and the given condition. The absolute value of a number a, denoted as |a|, is the distance of a from zero on the number line, regardless of direction. Therefore, if a is positive, |a|=a, and if a is negative, |a|=-a.

Given that z<5, both z-6 and z-5 are negative because z is less than both 5 and 6. Hence, we rewrite |z-6| as -(z-6) and |z-5| as -(z-5).

The original expression |z-6|-|z-5| becomes:

- (z - 6) - (-(z - 5)) = -z + 6 - (-z + 5) = -z + 6 + z - 5

The z terms cancel out, simplifying the expression to: 1

Answer 3

The expression |z-6|-|z-5|, when z<5, simplifies to 1 by considering the definition of absolute value and the given condition that z is less than 5.

Step-by-step explanation:

To rewrite the expression without the absolute value signs for z<5, we need to consider the definition of absolute value and the given condition. The absolute value of a number a, denoted as |a|, is the distance of a from zero on the number line, regardless of direction. Therefore, if a is positive, |a|=a, and if a is negative, |a|=-a.

Given that z<5, both z-6 and z-5 are negative because z is less than both 5 and 6. Hence, we rewrite |z-6| as -(z-6) and |z-5| as -(z-5).

The original expression |z-6|-|z-5| becomes:

- (z - 6) - (-(z - 5)) = -z + 6 - (-z + 5) = -z + 6 + z - 5

The z terms cancel out, simplifying the expression to: 1

Answer 4

The expression |z-6|-|z-5|, when z<5, simplifies to 1 by considering the definition of absolute value and the given condition that z is less than 5.

Step-by-step explanation:

To rewrite the expression without the absolute value signs for z<5, we need to consider the definition of absolute value and the given condition. The absolute value of a number a, denoted as |a|, is the distance of a from zero on the number line, regardless of direction. Therefore, if a is positive, |a|=a, and if a is negative, |a|=-a.

Given that z<5, both z-6 and z-5 are negative because z is less than both 5 and 6. Hence, we rewrite |z-6| as -(z-6) and |z-5| as -(z-5).

The original expression |z-6|-|z-5| becomes:

- (z - 6) - (-(z - 5)) = -z + 6 - (-z + 5) = -z + 6 + z - 5

The z terms cancel out, simplifying the expression to: 1

Answer 5

The expression |z-6|-|z-5|, when z<5, simplifies to 1 by considering the definition of absolute value and the given condition that z is less than 5.

Step-by-step explanation:

To rewrite the expression without the absolute value signs for z<5, we need to consider the definition of absolute value and the given condition. The absolute value of a number a, denoted as |a|, is the distance of a from zero on the number line, regardless of direction. Therefore, if a is positive, |a|=a, and if a is negative, |a|=-a.

Given that z<5, both z-6 and z-5 are negative because z is less than both 5 and 6. Hence, we rewrite |z-6| as -(z-6) and |z-5| as -(z-5).

The original expression |z-6|-|z-5| becomes:

- (z - 6) - (-(z - 5)) = -z + 6 - (-z + 5) = -z + 6 + z - 5

The z terms cancel out, simplifying the expression to: 1