# Appendix A - Equations and Inequalities - Page 418: 82

$(-\infty,1]$

#### 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 $(-\infty,1]$.

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