Question

solve the linear inequality for z. graph your answer in the number line. then write the solution in inequality notation.

z > 5(-4z + 2) - 8 + 19z

show your work here

hint: to add inequalities (<, >, ≤, ≥), type \less\ or \greater\

Explanation:

Step1: Expand the right - hand side

$$\begin{align*} z&>5(-4z + 2)-8 + 19z\\ z&>-20z+10 - 8+19z \end{align*}$$

Step2: Combine like terms on the right - hand side

$$\begin{align*} z&>(-20z + 19z)+(10 - 8)\\ z&>-z + 2 \end{align*}$$

Step3: Add \(z\) to both sides

$$\begin{align*} z+z&>-z + 2+z\\ 2z&>2 \end{align*}$$

Step4: Divide both sides by 2

$$\begin{align*} \frac{2z}{2}&>\frac{2}{2}\\ z&>1 \end{align*}$$

Answer:

The solution in inequality notation is \(z > 1\). On the number line, we have an open circle at \(z = 1\) (since \(z = 1\) is not included in the solution) and the line extends to the right of \(1\) towards positive infinity.