Question

1. a. solve the given inequality. b. describe the solution - set using set - builder notation. c. graph the solution set on a number line. 4(3b - 5) < - 31+13b

Explanation:

Step1: Simplify the left - hand side

First, simplify \(4(3b - 5)\) using the distributive property \(a(b - c)=ab - ac\). So \(4(3b - 5)=12b-20\). The inequality becomes \(12b - 20<-31 + 13b\).

Step2: Isolate the variable \(b\)

Subtract \(12b\) from both sides of the inequality: \((12b-20)-12b<(-31 + 13b)-12b\). This simplifies to \(- 20<-31 + b\).

Step3: Solve for \(b\)

Add 31 to both sides of the inequality: \(-20 + 31<-31 + b+31\). So \(11 < b\) or \(b>11\).

Step4: Set - builder notation

The set - builder notation for the solution set is \(\{b|b > 11,b\in\mathbb{R}\}\), where \(\mathbb{R}\) represents the set of all real numbers.

Step5: Graph on number line

Draw a number line. Mark a open - circle at 11 (since \(b\) is strictly greater than 11, not equal to 11) and shade the line to the right of 11.

Answer:

a. Solution: \(b>11\)
b. Set - builder notation: \(\{b|b > 11,b\in\mathbb{R}\}\)
c. On the number line, mark an open - circle at 11 and shade to the right.