Solution of inequalities

Q# 1: Find the solution set of 3x+1<7 ,$\forall x\in W$ and show on the number line.

Solution:

• First understand the meaning of $\forall x\in W$ $\forall$= For all values $\in$= belongs to W= whole numbers So it is basically saying the inequality is only for all values of x that belongs to whole numbers. So the solution set will have only whole numbers.

• Now Write the Inequality

  3x+1<7

• Solve the constant

  3x<7-1

  3x<6

• Divide by the coefficient of the variable

  \frac{3x}{3} < \frac{6}{3}

  x<2

• Our solution will only contains whole numbers and the answer is suggesting that all values less than 2. Therefore, the solution set is $\{0,1\}$ and represented in number line as:

Q# 2: Find the solution set of -6<2x+1<11, \forall x\in Z and show on the number line.

Solution:

• First understand the meaning of $\forall x\in \Z$ $\forall$= For all values $\in$= belongs to Z= Integers So it is basically saying the inequality is only for all values of x that belongs to integers. So the solution set will have only integer numbers.

• Now Write the Inequality

  -6<2x+1<11

• Since there are two inequalities, it can be expressed as

  -6<2x+1 and 2x+1<11

• Solve the constant

  -6-1<2x and 2x<11-1

  -7<2x and 2x<10

• Divide by the coefficient of the variable

  \frac{-7}{2}<\frac{2x}{2} and \frac{2x}{2}<\frac{10}{2}

  \frac{-7}{2}<x and x<5

  -3.5<x and x<5

• Our solution will only contains integer numbers and the answer is suggesting that all values greater than -3.5 and less than 5. Therefore, the solution set is $\{-3,-2,-1,0,1,2,3,4\}$ and represented in number line as: