Question

Translate the following phase into an inequality: “ the sum of 4x and 30 exceeds 14x. “ A) Inequality? B) Solve the inequality for x. C) Express the solution interval notation.

Answer 1

A.) Translate into inequality

`\begin{gathered} \text{The sum of 4x and 30 translates to} \\ 4x+30 \\ \\ \text{exceeds 14x translates to} \\ >14x \\ \\ \text{putting it together, the inequality is} \\ 4x+30>14x \end{gathered}`

B.) Solve for inequality x

`\begin{gathered} 4x+30>14x \\ 4x-14x>-30 \\ -10x>-30 \\ \\ \text{Divide both sides by }-10 \\ -10x>-30 \\ (-10x)/(-10)>(-30)/(-10) \\ \\ \text{Dividing negative in both sides flips the inequality} \\ x<3 \\ \\ \text{Therefore, the solution to the inequality is} \\ x<3 \end{gathered}`

C.) Express the solution in interval notation

The solution extends from negative infinity up to but not including 3, therefore, when expressed in interval notation, the solution is

`(-\infty,3)`