Solve the following inequalities

1. $3 x+1 \geqslant-2$
2. $y \geqslant 1$ and $-2 y \geqslant-2$
3. $y^{2}\left(x^{2}+1\right)-1 \leqslant x^{2}$
## We correct:

1. We have

$$
\begin{aligned}
& 3 x+1 \geqslant-2 \\
\Leftrightarrow & 3 x \geqslant-3 \\
\Leftrightarrow & x \geqslant-1 .
\end{aligned}
$$

2. We know that $-2 y \geqslant-2 \Leftrightarrow y \leqslant 1$ so we have $y \leqslant 1$ and $y \geqslant 1$ hence $y=1$.
3. We have

$$
\begin{aligned}
y\left(x^{2}+1\right) & -1 \leqslant x^{2} \\
\Leftrightarrow y\left(x^{2}+1\right) & \leqslant x^{2}+1
\end{aligned}
$$

We know that $x^{2} \geqslant 0$ so $x^{2}+1>0$ and we can divide both sides

$$
\begin{aligned}
& y\left(x^{2}+1\right) \leqslant x^{2}+1 \\
\Leftrightarrow & y \leqslant 1
\end{aligned}
$$

## Remark 5.

There are several ways to solve an inequality. The first is to proceed by equivalence, which involves making a series of statements that start with the statement and end with the solution, all of which are equivalent to each other.

$$
\begin{aligned}
& \text { statement } \\
& \Leftrightarrow(1) \\
& \Leftrightarrow(2) \\
& \Leftrightarrow \cdots \\
& \Leftrightarrow(n) \\
& \Leftrightarrow \text { solution }
\end{aligned}
$$

The advantage here is that we are done as soon as we have obtained the solution. We will have shown both that it works and that these are the only correct ones. The disadvantage is that we can only use equivalences and not implications. The second way is to proceed by Analysis-Synthesis. This involves obtaining as much information as possible about the solution using implications. This step is called the analysis.

$$
\text { statement } \Rightarrow(1) \Rightarrow(2) \Rightarrow \cdots \Rightarrow(n) \Rightarrow \text { solution (?) }
$$

Once we think we have a good grasp of the solution, we need to verify that it works: solution $\Rightarrow$ statement. This is the synthesis.