Chapter 2.2, Problem 31E

To determine

Equivalent inequalities to the inequality $x-b<3$ among

  $x-b-3<0$

  $0>b-x+3$

  $x<3-b$

  $-3<b-x$

Answer to Problem 31E

The inequalities $x-b-3<0$ and $0>b-x+3$ are equivalent to the inequality $x-b<3$

Explanation of Solution

Given:

The inequalities are

  $x-b<\mathrm{3...........................}(1)$

  $x-b-3<\mathrm{0.............................}(A)$

  $0>b-x+\mathrm{3.....................................}(B)$

  $x<3-b\mathrm{...................................}(C)$

  $-3<b-x\mathrm{.............................}(D)$

Calculation:

Solving inequality (A)

  $\begin{array}{l}x-b-3<0\\ \Rightarrow x-b-3+3<0+3\\ \Rightarrow x-b<3\end{array}$

Solving inequality (B)

  $\begin{array}{l}0>b-x+3\\ \Rightarrow 0+x>b-x+3+x\\ \Rightarrow x>b+3\\ \Rightarrow x-b<b+3-b\\ \Rightarrow x-b<3\end{array}$

Solving inequality (C)

  $\begin{array}{l}x<3-b\\ \Rightarrow x+b<3-b+b\\ \Rightarrow x+b<3\end{array}$

Solving inequality (D)

  $\begin{array}{l}-3<b-x\\ \Rightarrow -3+x<b-x+x\\ \Rightarrow x-3<b\end{array}$

  $\begin{array}{l}\Rightarrow x-3-b>b-b\\ \Rightarrow x-3-b>0\\ \Rightarrow x-3-b+3>0+3\\ \Rightarrow x-b>3\end{array}$

From the solutions we see that inequalities (A) and (B) are equivalent to the inequality $x-b<3$

Thus, the inequalities $x-b-3<0$ and $0>b-x+3$ are equivalent to the inequality $x-b<3$