Questions: Solve the absolute value inequality. Graph the solution set. 1-2x-6<-1 Select the correct choice and, if necessary, fill in the answer box within your choice. A. The solution set in interval notation is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. The solution is the empty set. Graph the solution set. Choose the correct answer. A. B. C. D. E. F.

$Solve the absolute value inequality. Graph the solution set. 1-2x-6<-1 Select the correct choice and, if necessary, fill in the answer box within your choice. A. The solution set in interval notation is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. The solution is the empty set. Graph the solution set. Choose the correct answer. A. B. C. D. E. F.$

Transcript text: Solve the absolute value inequality. Graph the solution set. \[ |1-2 x|-6<-1 \] Select the correct choice and, if necessary, fill in the answer box within your choice. A. The solution set in interval notation is $\square$ (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. The solution is the empty set. Graph the solution set. Choose the correct answer. A. B. c. D. E. F.

Solution

Solution Steps

Step 1: Simplify the Inequality

Given the inequality: \[ |1 - 2x| - 6 < -1 \]

First, isolate the absolute value expression by adding 6 to both sides: \[ |1 - 2x| - 6 + 6 < -1 + 6 \] \[ |1 - 2x| < 5 \]

Step 2: Solve the Absolute Value Inequality

The inequality $ |1 - 2x| < 5 $ can be rewritten as a compound inequality: \[ -5 < 1 - 2x < 5 \]

Step 3: Solve the Compound Inequality

Break the compound inequality into two parts and solve for $ x $:

$ -5 < 1 - 2x $ \[ -5 - 1 < -2x \] \[ -6 < -2x \] \[ \frac{-6}{-2} > x \] \[ 3 > x \] \[ x < 3 \]
$ 1 - 2x < 5 $ \[ 1 - 5 < 2x \] \[ -4 < 2x \] \[ \frac{-4}{2} < x \] \[ -2 < x \]

Combining these results, we get: \[ -2 < x < 3 \]