Question

solve the inequality $x + 3 \leq -1$.
$x + 3 \leq -1$
$\\ \\ -3\\ \\ \\ \\ -3$
$\\ \\ \\ \\ x \leq -4$
you subtracted the same value from both sides of the inequality. that means $x \leq -4$ is the solution set of $x + 3 \leq -1$.
test a value not included in the solution set.
$-3 \leq -4$ false
$x + 3 \leq -1$
$-3 + 3 \leq -1$
$0 \leq -1$
true
false
?
test another value incl
$-5 \leq -4$ true
$x + 3 \leq -1$
$-5 + 3 \leq -1$
$-2 \leq -1$
sub

Explanation:

Step1: Analyze the inequality

We have the inequality \(x + 3\leq - 1\). To solve for \(x\), we need to isolate \(x\) on one side of the inequality.

Step2: Subtract 3 from both sides

According to the properties of inequalities, if we subtract the same number from both sides of an inequality, the direction of the inequality sign remains the same. So we subtract 3 from both sides of \(x + 3\leq - 1\):
\[

$$\begin{align*} x+3 - 3&\leq-1 - 3\\ x&\leq - 4 \end{align*}$$

\]

Step3: Test the solution (optional verification)

• For a value not in the solution set, say \(x=-3\) (since \(-3>-4\)), substitute into the original inequality: \(-3 + 3=0\), and \(0\leq - 1\) is false, which is consistent with \(x = - 3\) not being in the solution set.
• For a value in the solution set, say \(x = - 5\) (since \(-5\leq-4\)), substitute into the original inequality: \(-5 + 3=-2\), and \(-2\leq - 1\) is true, which is consistent with \(x=-5\) being in the solution set.

Answer:

The solution to the inequality \(x + 3\leq - 1\) is \(x\leq - 4\)