Question

10 match each inequality to the number line that represents its solutions.
a. ( |x - 3|>1 )
b. ( |x - 3|<1 )
c. ( |x - 1|>3 )
d. ( |x - 1|<3 )

Explanation:

Step1: Solve inequality \(|x - 3|>1\)

By the definition of absolute value, \(x - 3>1\) or \(x - 3<-1\).
For \(x - 3>1\), add \(3\) to both sides: \(x>4\).
For \(x - 3<-1\), add \(3\) to both sides: \(x<2\).

Step2: Solve inequality \(|x - 3|<1\)

By the definition of absolute value, \(-1Add \(3\) to all parts: \(2

Step3: Solve inequality \(|x - 1|>3\)

By the definition of absolute value, \(x - 1>3\) or \(x - 1<-3\).
For \(x - 1>3\), add \(1\) to both sides: \(x>4\).
For \(x - 1<-3\), add \(1\) to both sides: \(x<-2\).

Step4: Solve inequality \(|x - 1|<3\)

By the definition of absolute value, \(-3Add \(1\) to all parts: \(-2

Answer:

a. The number line with open - circles at \(2\) and \(4\) (the first number line).
b. The number line with open - circles at \(2\) and \(4\) (the third number line).
c. The number line with open - circles at \(-2\) and \(4\) (the second number line).
d. The number line with open - circles at \(-2\) and \(4\) (the fourth number line).