Question

2. Solve the inequality for x. Show each step of the solution. 16x > 12(2x – 6) – 24

Answer 1

Simplifying 2x + 6 = 4x + -2 Reorder the terms: 6 + 2x = 4x + -2 Reorder the terms: 6 + 2x = -2 + 4x Solving 6 + 2x = -2 + 4x Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4x' to each side of the equation. 6 + 2x + -4x = -2 + 4x + -4x Combine like terms: 2x + -4x = -2x 6 + -2x = -2 + 4x + -4x Combine like terms: 4x + -4x = 0 6 + -2x = -2 + 0 6 + -2x = -2 Add '-6' to each side of the equation. 6 + -6 + -2x = -2 + -6 Combine like terms: 6 + -6 = 0 0 + -2x = -2 + -6 -2x = -2 + -6 Combine like terms: -2 + -6 = -8 -2x = -8 Divide each side by '-2'. x = 4 Simplifying x = 4 the answer is x=4

Answer 2

Remark

The first thing you should do is to get a program to give you a graph. I use Desmos. We'll talk about the solution shown by the graph when we are done with the solution

Solution

Remove the brackets on the right

16x > 12*2x - 12*6 - 24

16x > 24x - 72 - 24 Combine like terms

16x > 24x - 96 Add 96 to both sides.

16x + 96 > 24x Subtract 16x from both sides.

96 > 24x - 16x

96 > 8x Divide 8

96/8 > x

12 > x

Graph

Now we can discuss the graph. The dotted broken line which separates the white from the pink is x = 12. That's the boundary to this inequality. The line itself is parallel to the y axis. Everything to the left of x = 12 is a possible solution to the inequality if it is a number on the x axis. What to do with the y values is an answer for another day. There are many solutions to this inequality, unlike an equation which has only one at the level of your current math class.