Solution: Step:1 Transform into a single fraction. Step:1.1 Express 1 as a fraction by creating a common denominator: frac1 - x1 - x - frac11 - x Step:1.2 Merge the numerators while keeping the common denominator: frac(1 - x) - 11 - x Step:2 Streamline the numerator. Step:2.1 Perform the subtraction of 1 from 1: frac0 - x1 - x Step:2.2 Combine the terms - x and 0: frac- x1 - x Step:3 Reposition the negative sign to the front of the fraction: - fracx1 - x Knowledge Notes: The problem at hand involves simplifying a rational expression. The process of simplification typically includes combining like terms, finding common denominators, and reducing fractions to their simplest form. Here are the relevant knowledge points: 1. Common Denominator: To combine fractions, they must have the same denominator. A common denominator is found by ensuring that each term of the expression has the same bottom part (denominator). 2. Combining Numerators: Once a common denominator is established, the numerators (top parts) of the fractions can be combined by performing the indicated operations (addition or subtraction). 3. Simplifying the Numerator: After combining the numerators, the resulting expression in the numerator can often be simplified by combining like terms or canceling out terms. 4. Negative Sign in Fractions: A negative sign can be placed in front of the fraction, in the numerator, or in the denominator without changing the value of the fraction. It is often moved in front of the fraction for clarity. The process of simplifying the given expression, 1 - frac11 - x, involves these steps to arrive at the simplified form, - fracx1 - x.

Simplify 1-1/(1-x)

The question is asking to take the algebraic expression 1-1/(1-x) and rewrite it in a simpler or more reduced form. This often involves manipulating the expression to consolidate terms, eliminate fractions, or reorganize the equation to present it in a more straightforward manner without changing its value or meaning. The process may involve applying algebraic rules such as combining like terms, finding common denominators, and simplifying fractions.

$1 - \frac{1}{1 - x}$

Answer

Expert–verified

Solution:

Step:1

Transform into a single fraction.

Step:1.1

Express $1$ as a fraction by creating a common denominator: $\frac{1 - x}{1 - x} - \frac{1}{1 - x}$

Step:1.2

Merge the numerators while keeping the common denominator: $\frac{(1 - x) - 1}{1 - x}$

Step:2

Streamline the numerator.

Step:2.1

Perform the subtraction of $1$ from $1$: $\frac{0 - x}{1 - x}$

Step:2.2

Combine the terms $- x$ and $0$: $\frac{- x}{1 - x}$

Step:3

Reposition the negative sign to the front of the fraction: $- \frac{x}{1 - x}$

Knowledge Notes:

The problem at hand involves simplifying a rational expression. The process of simplification typically includes combining like terms, finding common denominators, and reducing fractions to their simplest form. Here are the relevant knowledge points:

Common Denominator: To combine fractions, they must have the same denominator. A common denominator is found by ensuring that each term of the expression has the same bottom part (denominator).
Combining Numerators: Once a common denominator is established, the numerators (top parts) of the fractions can be combined by performing the indicated operations (addition or subtraction).
Simplifying the Numerator: After combining the numerators, the resulting expression in the numerator can often be simplified by combining like terms or canceling out terms.
Negative Sign in Fractions: A negative sign can be placed in front of the fraction, in the numerator, or in the denominator without changing the value of the fraction. It is often moved in front of the fraction for clarity.

The process of simplifying the given expression, $1 - \frac{1}{1 - x}$, involves these steps to arrive at the simplified form, $- \frac{x}{1 - x}$.