Solve the inequality. Write the solution in set-builder notation. - 2(9x + 6) > - 12 - 18x Solve the inequality. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A (Type an inequality.) OB. The solution is all real numbers. c. There is no solution. Write the solution in set-builder notation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. \ x | Box\ (Type an inequality.) O B. \ x |x is an element of the set of all real numbers} O C. \ x |x is an element of the empty set}
To solve the inequality, we can start by distributing -2 to the terms inside the parentheses:
-18x - 12 > -12 - 18x
Next, simplify both sides of the inequality:
-18x - 12 > -12 - 18x
-18x + 18x > -12 + 12
0 > 0
Since 0 is not greater than 0, the inequality is false. Therefore, there is no solution to the inequality.
The correct answer is: c. There is no solution.
In set-builder notation, we can represent this solution as:
\ x | x is an element of the empty set} (no elements satisfy the inequality)
Let's solve the inequality step-by-step:
-2(9x + 6) > -12 - 18x
First, distribute -2:
-18x - 12 > -12 - 18x
Next, combine like terms:
-18x + 18x - 12 > -12
The -18x and +18x cancel each other out, leaving:
-12 > -12
Since -12 is not greater than -12, this inequality is not true.
Therefore, the correct choice is:
C. There is no solution.
And the solution in set-builder notation is:
\(\emptyset\)
To solve the inequality -2(9x + 6) > -12 - 18x, we can start by simplifying both sides of the inequality.
-2(9x + 6) > -12 - 18x
Distribute -2 on the left side:
-18x - 12 > -12 - 18x
Next, we can simplify and combine like terms:
-18x + 18x > -12 - 12
0 > -24
At this point, we have 0 > -24, which is a true statement. This means that the inequality is always true, regardless of the value of x. Therefore, the solution is all real numbers.
In set-builder notation, we can represent this solution as:
{x | x is an element of the set of all real numbers} (Choice B)