Question

solve the inequality and graph the solution.
2(z + 2) + 12 < 8
plot the endpoints. select an endpoint to change it from closed to open. select the middle of
the segment, ray, or line to delete it.
← │ │ │ │ │ │ │ │ │ │ →
-5 -4 -3 -2 -1 0 1 2 3 4 5

Explanation:

Step1: Expand the left side

First, we expand \(2(z + 2)\) using the distributive property \(a(b + c)=ab+ac\). So \(2(z + 2)=2z+4\). Then the inequality becomes \(2z + 4+12\lt8\).

Step2: Combine like terms

Combine the constant terms on the left side: \(4 + 12 = 16\). So the inequality is now \(2z+16\lt8\).

Step3: Subtract 16 from both sides

Subtract 16 from both sides of the inequality: \(2z+16 - 16\lt8 - 16\). This simplifies to \(2z\lt - 8\).

Step4: Divide both sides by 2

Divide both sides of the inequality by 2: \(\frac{2z}{2}\lt\frac{-8}{2}\). Which gives \(z\lt - 4\).

Answer:

The solution to the inequality \(2(z + 2)+12\lt8\) is \(z\lt - 4\). To graph this, we draw an open circle at \(z=-4\) (since the inequality is strict, \(z
eq - 4\)) and draw a ray pointing to the left (towards negative infinity) from the open circle at \(z = - 4\).