Find the solution set to each inequality. Express the solution in set notation

Accepted Solution

The stategy is always the same: you have to manipulate the inequality so that all terms involving the variable sit on one side of the inequality and al constant terms lie on the other.Then, you gather like terms and you divide by the variable coefficient to solve for the variable.1) Start with[tex]6m+2<5m-4[/tex]Subtract 5m and 2 from both sides:[tex]m<-6[/tex]In set notation,[tex]\{m \in \mathbb{R}:\ m<-6\}[/tex]2) Start with[tex]\dfrac{a}{5}+8\leq 13[/tex]Subtract 8 from both sides:[tex]\dfrac{a}{5}\leq 5[/tex]Multiply both sides by 5:[tex]a\leq 25[/tex]In set notation,[tex]\{a \in \mathbb{R}:\ a\leq 25\}[/tex]3) Start with[tex]-3(x-7)>-2[/tex]Distribute the -3:[tex]-3x+21>-2[/tex]Subtract 21 from both sides:[tex]-3x>-23[/tex]Divide both sides by -3. Since we're dividing by a negative number, we have to switch the inequality sign:[tex]x<\dfrac{23}{3}[/tex]In set notation,[tex]\left\{x \in \mathbb{R}:\ x<\dfrac{23}{3}\right\}[/tex]4) Start with[tex]8(p-6)>4(p-4)[/tex]Distribute both sides:[tex]8p-48>4p-16[/tex]Subtract 4p from both sides[tex]4p-48>-16[/tex]Add 48 to both sides:[tex]4p>32[/tex]Divide both sides by 4:[tex]p>8[/tex]In set notation,[tex]\{p \in \mathbb{R}:\ p>8\}[/tex]