How to add rational expressions.

First, some general rules to remember:

Rational expressions (fractions) can only be added or subtracted if they have a common denominator.
The numerator and denominator of a fraction may be multiplied by the same quantity. This will result in a fraction that is equivalent to the original fraction.
For a fractional answer to be in final form, the fraction must be reduced to lowest terms.
Adding or subtracting rational expressions is a four-step process:

Write all fractions as equivalent fractions with a common denominator.
Combine the fractions as a single fraction that has the common denominator.
Simplify the expression in the top of the fraction.
Reduce the fraction to lowest terms.
To see this process, we'll look at some examples.

Example 1

Step 1: To find a common denominator, start by factoring the denominators of both fractions.

The least common denominator of these two fractions is 2(x+6)(x-6).

Looking at the first fraction, we can see that the factor missing from its denominator is (x-6). In the next step, we'll multiply the top and bottom of the first fraction by (x-6).

Looking at the second fraction, we can see that the factor missing from its denominator is 2. In the next step, we'll multiply the top and bottom of the second fraction by 2.

Step 2: Now that the fractions have the same denominators, we'll create one combined fraction by adding the two fractions together:

Step 3: We'll simplify the top of the fraction. Notice that we are only simplifying the top of the fraction; the bottom of the fraction remains unchanged.

Step 4: Now we'll reduce the fraction to lowest terms

Example 2

Step 1: To find a common denominator, start by factoring the denominators of both fractions.

The least common denominator of these two fractions is (x+4)(x+1)(x+3).

Looking at the first fraction, we can see that the factor missing from its denominator is (x+3). In the next step, we'll multiply the top and bottom of the first fraction by (x+3).

Looking at the first fraction, we can see that the factor missing from its denominator is (x+4). In the next step, we'll multiply the top and bottom of the first fraction by (x+4).

Step 2: Now that the fractions have the same denominators, we'll create one combined fraction by adding the two fractions together:

Step 3: We'll simplify the top of the fraction. Notice that we are only simplifying the top of the fraction; the bottom of the fraction remains unchanged.

Step 4: Now we'll reduce the fraction to lowest terms

and y≤ -x +4 .

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

.

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

y≤ -x +4 .