Question

solve the inequality -x + 2 < -3x + 8. graph the solution on a number - line and represent the solution in interval - notation when possible. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution is x < 3 (simplify your answer. type an inequality.) b. the solution is all real numbers c. there is no solution. choose the correct graph of the inequality.

Explanation:

Step1: Solve the inequality

Starting with \(-x + 2< - 3x+8\). Add \(3x\) to both sides: \(-x+3x + 2< - 3x+3x + 8\), which simplifies to \(2x+2 < 8\). Then subtract 2 from both sides: \(2x+2 - 2<8 - 2\), getting \(2x<6\). Divide both sides by 2: \(\frac{2x}{2}<\frac{6}{2}\), so \(x < 3\).

Step2: Analyze the graph representation

For the inequality \(x < 3\), on a number - line, we have an open circle at \(x = 3\) (because \(x\) is strictly less than 3, not equal to 3) and the line extends to the left.

Answer:

The solution is \(x < 3\). The correct graph is one with an open - circle at 3 and the line extending to the left. Among the given options, if there is an option with an open - circle at 3 and the line going left, that is the correct graph. Without seeing the specific details of each graph option clearly, we know the solution in interval notation is \((-\infty,3)\).