Animated Solution for Mathematics - Basic Mathematics: The number of elements in the set $x\in R:(|x|-3)|x+4|=6$ is equal to

To solve the equation $(|x|-3)|x+4|=6$, we must identify the "hinges" where the expressions inside the modulus bars change sign. For $|x|$, the hinge is at $x=0$. For $|x+4|$, the hinge is at $x=-4$.

These points divide the number line into three distinct intervals: 1. $x<-4$ 2. $-4\le x<0$ 3. $x\ge 0$

We will analyze the behavior of the function within each of these regions to find the valid intersection points.

In this region, both $x$ and $x+4$ are non-negative. The modulus bars can be removed without changing the signs of the terms.

The equation simplifies to:

Expanding the terms, we obtain:

Using the quadratic formula, we find:

Since we assumed $x\ge 0$, we reject the negative root. The valid solution in this region is:

In this region, $x$ is negative, so $|x|=-x$. However, $x+4$ remains non-negative, so $|x+4|=x+4$.

The equation transforms into:

Factoring out the negative sign, we get:

Checking the discriminant $D=b^2-4ac$:

Because the discriminant is negative, there are no real solutions in this interval.

In this region, both $x$ and $x+4$ are negative. Thus, $|x|=-x$ and $|x+4|=-(x+4)$.

The equation becomes:

The two negative signs cancel out, simplifying the expression to:

Factoring the quadratic equation:

This yields potential roots $x=-6$ and $x=-1$. Given our condition $x<-4$, we reject $x=-1$ as it falls outside the interval. The valid solution is $x=-6$.

By exploring all three regions, we have identified the valid intersection points. The equation $(|x|-3)|x+4|=6$ is satisfied by exactly two values: