Question

solve the following inequality. write the solution set using interval notation.\\(\frac{-x + 2}{2}-\frac{1 - 7x}{8}<-10\\)\\(\\)select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\\)\\(\circ\\) a. the solution set is \\(\square\\).\\(\\) (type your answer in interval notation. use integers or fractions for any numbers in the expression.)\\(\\)\\(\circ\\) b. the solution set is \\(\varnothing\\).

Explanation:

Step1: Eliminate denominators

Multiply each term by 8 (the least common multiple of 2 and 8) to clear the fractions:
$$8\times\frac{-x + 2}{2}-8\times\frac{1 - 7x}{8}<8\times(-10)$$
Simplify each term:
$$4(-x + 2)-(1 - 7x)<-80$$

Step2: Distribute and simplify

Distribute the 4 and the negative sign:
$$-4x + 8 - 1 + 7x<-80$$
Combine like terms:
$$3x + 7<-80$$

Step3: Isolate the variable

Subtract 7 from both sides:
$$3x<-80 - 7$$
$$3x<-87$$
Divide both sides by 3:
$$x<-29$$

Answer:

A. The solution set is $(-\infty, -29)$