Question

Solve the inequality. $$ 2 x+3<-1 \text { or } 3 x-5>-2 $$

Short answer

The solution is \(x<-2\) or \(x>1\).

Step-by-step solution

Solving the first inequality

The first inequality \(2x+3<-1\) should be solved by isolating the variable x. The first step is subtracting 3 from both sides: \(2x<-1-3\), which simplifies to \(2x<-4\). Then, it's divided by 2: \(x<-4/2\), resulting in \(x<-2\).

Solving the second inequality

The second inequality \(3x-5>-2\) needs to be solved similarly. The first step is adding 5 to both sides: \(3x>-2+5\), which simplifies to \(3x>3\). Then it's divided by 3: \(x>3/3\), yielding \(x>1\).

Finding the solution set

The compound inequality states an 'or' condition, so the solution set is the union of the solutions of both inequalities, which are \(x<-2\) or \(x>1\).