3. 3x+3leqslant x-1 and 1+xgt 3-x Inequality: __ Graph: Interval Notation: __

To solve the system of inequalities, we will solve each inequality separately and then find the intersection of the solutions.<br /><br />1. Solving the first inequality: $3x+3\leqslant x-1$<br /><br />Subtract $x$ from both sides:<br />$2x+3\leqslant -1$<br /><br />Subtract 3 from both sides:<br />$2x\leqslant -4$<br /><br />Divide both sides by 2:<br />$x\leqslant -2$<br /><br />2. Solving the second inequality: $1+x\gt 3-x$<br /><br />Add $x$ to both sides:<br />$1+2x\gt 3$<br /><br />Subtract 1 from both sides:<br />$2x\gt 2$<br /><br />Divide both sides by 2:<br />$x\gt 1$<br /><br />Now, we need to find the intersection of the solutions for both inequalities. The solution for the first inequality is $x\leqslant -2$, and the solution for the second inequality is $x\gt 1$. Since there is no overlap between these two intervals, there is no solution to the system of inequalities.<br /><br />Therefore, the inequality is: $x\leqslant -2$ or $x\gt 1$<br /><br />Graph: The graph would show two separate regions, one for $x\leqslant -2$ and one for $x\gt 1$, with no overlapping region.<br /><br />Interval Notation: The interval notation for this system of inequalities is $(-\infty, -2]$ or $(1, \infty)$.