Answer
$(-\infty,1]$
Work Step by Step
Step 1: $-4x+3\geq-2+x$
Step 2: Adding $2$ to both sides, $-4x+3+2\geq-2+x+2$
Step 3: $-4x+5\geq x$
Step 4: Subtracting $x$ from both sides, $-4x+5-x\geq x-x$
Step 5: $-5x+5\geq0$
Step 6: Subtracting 5 from both sides, $-5x+5-5\geq0-5$
Step 7: $-5x\geq-5$
Step 8: Dividing both sides by -5 (this reverses the direction of the inequality symbol):
$\frac{-5x}{-5} \leq \frac{-5}{-5}$
Step 9: $x\leq1$
According to the inequality, the interval includes $1$ and all values less than $1$. Since $1$ is part of the interval, a square bracket is used on its side. On the other hand, we represent all values less than $1$ by the symbol $-\infty$. Therefore, a parenthesis is used on its side as parentheses are always used wherever $\infty$ is used.
Therefore, the interval notation for this inequality is written as $(-\infty,1]$.
