Question

Solve the inequality and graph the solution. $$ -5<3 x+4<19 $$

Short answer

The solution to the inequality \(-5 < 3x+4 < 19\) is \(-3 < x < 5\) and graphically, this can be represented as a line segment between -3 and 5 on the number line, with open points at -3 and 5.

Step-by-step solution

Isolate x in each inequality

Take the inequality \(-5 < 3x+4 < 19\). The first step will be to eliminate the constant on the side with \(x\) by subtracting 4 from all expressions. This results in the inequality \(-5 - 4 < 3x < 19 - 4\), which simplifies to \(-9 < 3x < 15\).

Further Isolate x in each inequality

Now, divide all parts by 3 to leave \(x\) on its own. This will result in \(-9/3 < x < 15/3\), simplifying to \(-3 < x < 5\). This is the solution to the compound inequality.

Graph the solution on a number line

To graph the solution, draw a number line. Mark the points -3 and 5 on it. These are open points since -3 and 5 aren't part of the solution. The interval between -3 and 5 is the solution, so shade this section on the number line.