**Solving the Inequality x/(x+4) <=>

To solve the inequality x/(x+4) <= 1,="" we="" need="" to="" find="" the="" values="" of="" x="" that="" satisfy="" the="" inequality.="" let's="" go="" through="" the="" steps="" to="" solve="">

**Step 1: Find the domain**
The first step is to determine the domain of the expression x/(x+4). The expression is defined for all real numbers except when the denominator (x+4) is equal to zero. So, x cannot be equal to -4.

**Step 2: Solve the equation**
To solve the equation x/(x+4) = 1, we can multiply both sides by (x+4) to eliminate the denominator:

x = (x+4)

Expanding the equation, we get:

x = x + 4

Subtracting x from both sides, we have:

0 = 4

This is a contradiction, which means the equation x/(x+4) = 1 has no solution.

**Step 3: Set up a number line**
Since the equation has no solution, we need to look for the values of x that make the inequality x/(x+4) <= 1="" true.="" to="" do="" this,="" we="" set="" up="" a="" number="" line="" and="" mark="" the="" critical="" points,="" which="" are="" the="" values="" that="" make="" the="" inequality="">

-4

**Step 4: Test the intervals**
Now, we test the intervals on the number line to determine which intervals satisfy the inequality.

For x < />
Let's take a value x = -5. Plugging it into the inequality, we have:

(-5)/((-5)+4) <=>

Simplifying, we get:

-5/-1 <=>

This is true, so the interval x < -4="" satisfies="" the="" />

For x > -4:
Let's take a value x = 0. Plugging it into the inequality, we have:

0/(0+4) <=>

Simplifying, we get:

0/4 <=>

This is also true, so the interval x > -4 satisfies the inequality.

**Step 5: Write the solution**
Combining the intervals, we can write the solution to the inequality as:

x < -4="" or="" x="" /> -4

In interval notation, the solution is (-∞, -4) U (-4, ∞).

Therefore, the values of x that satisfy the inequality x/(x+4) <= 1="" are="" all="" real="" numbers="" except="" x="-4." 1="" are="" all="" real="" numbers="" except="" x="">