In this problem, we want to solve two inequalities.
There is one rule that is important to remember with inequalities:
- If you multiply or divide by a negative number in the last step, you must flip the inequality symbols.
We are given:
[tex]2x>-10\text{ and }3x<24[/tex]We can solve these one at a time. Beginning with the first inequality,
[tex]2x>-10[/tex]Divide by 2 on both sides:
[tex]\begin{gathered} \frac{2x}{x}>\frac{-10}{2} \\ \\ x>-5 \end{gathered}[/tex]Moving on to the second inequality, we get:
[tex]\begin{gathered} 3x<24 \\ \\ \text{ Divide by 3 on both sides:} \\ \\ \frac{3x}{3}<\frac{24}{3} \\ \\ x<8 \end{gathered}[/tex]So now we know that x is greater than -5 but less than 8.
Let's see what that looks like on a numberline:
We see that the values of x can be included within a lower bound and an upper bound at -5 and 8.
This means we can write the final inequality as
[tex]\boxed{-5
