Solve using set builder notation, show all work. 12x-25 is greater than or = to 3x-5(x+6)
5 ( x + 6 ) = 5 x + 30
12 x - 25 ¡Ý 3 x - 5 ( x + 6 )
12 x - 25 ¡Ý 3 x - ( 5 / x - 6 ) )
12 x - 25 3 x - ( 5 x + 30 )
12 x - 25 ¡Ý 3 x - 5 x - 30
12 x - 25 ¡Ý - 2 x - 30
Add 2 x to both dides
12 x - 25 + 2 x ¡Ý - 2 x + 2 x - 30
14 x - 25 ¡Ý - 30 Add 25 to both sides
14 x - 25 + 25 ¡Ý - 30 + 25
14 x ¡Ý - 5 Divide both sides by 14
x ¡Ý - 5 / 4
Remark:
¡Ý
mean greater or equal
To solve the inequality 12x - 25 ≥ 3x - 5(x + 6), we need to simplify it step-by-step.
First, distribute the -5 to the terms inside the parentheses:
12x - 25 ≥ 3x - 5x - 30
Next, combine like terms:
12x - 25 ≥ -2x - 30
Now, let's isolate x. Move -2x to the left side by adding it to both sides of the inequality:
12x + 2x - 25 ≥ -2x + 2x - 30
Simplifying the left side:
14x - 25 ≥ -30
Now, move -25 to the right side by adding 25 to both sides of the inequality:
14x - 25 + 25 ≥ -30 + 25
Simplifying:
14x ≥ -5
To solve for x, divide both sides of the inequality by 14:
(1/14)(14x) ≥ (-5/14)
Simplifying:
x ≥ -5/14
Therefore, the solution in set-builder notation is:
{x | x ≥ -5/14}