Animated Solution for Mathematics - Basic Mathematics: Solve $|x^2+4x+3|+|2x+5|=0$

The equation provided is $|x^2+4x+3|+|2x+5|=0$.

The modulus function is a one-way mirror; for any real number $a$, the modulus $|a|$ is always greater than or equal to zero. This is the core geometric reality we must visualize.

We are adding two quantities, $|x^2+4x+3|$ and $|2x+5|$, and their sum is zero. Since both terms are non-negative, the only way their sum can be zero is if each term is zero individually.

This is the 'AND' condition. We are looking for a value that forces both terms to vanish simultaneously.

Let us break this down. First, we focus on the term $|x^2+4x+3|=0$.

This implies that $x^2+4x+3=0$. We can factorize it by splitting the middle term:

This gives us two potential candidates for $x$: $x=-3$ and $x=-1$. These are the only two points on the number line where the first part of our equation is satisfied.

Now, we turn our attention to the second term: $|2x+5|=0$.

This implies $2x+5=0$. Solving for $x$, we get $2x=-5$, which means:

This is the only point on the number line where the second part of our equation is satisfied. If $x$ is anything else, this term will be strictly positive.

Here is where the magic happens. We have two sets of solutions: the first set is $\{-3,-1\}$ and the second set is $\{-2.5\}$.

For the original equation to hold, we need a value of $x$ that exists in both sets. We are looking for the intersection:

As we look at these values on the number line, we see they are distinct. There is no overlap.

Because there is no single value of $x$ that can satisfy both conditions at the same time, we must conclude that there is no solution.