Chapter 8.7, Problem 21IP

To determine

Solve the inequality and check the solutions.

Answer to Problem 21IP

The solution of $14\frac{1}{2}<b-1\frac{1}{4}$ is $b>15\frac{3}{4}\text{or}(15\frac{3}{4},\infty )$

Explanation of Solution

Given:

The Inequality: $14\frac{1}{2}<b-1\frac{1}{4}$

Concept Used:

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. a ≠ b says that a is not equal to b. a < b says that a is less than b. a > b says that a is greater than b.

There are four different types of inequalities:

Greater than − $(>)$ ; Less than − $(<)$ ; Greater than or equal to − $(\ge )$ ; Less than or equal to − $(\le )$

For Inequality equation: If $b>c\Rightarrow c<b\text{or}b<c\Rightarrow c>b$

Rules for solving inequality equations:

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

But these things do change the direction of the inequality (" $<$ " becomes " $>$ " for example):

• Multiply (or divide) both sides by a negative number
• Swapping left and right hand sides
• If there is a parenthesis $\left(\right)$ on the interval, then we use the strict inequality symbols $<\text{ or}>$

If there is a bracket $\left[\right]$ on the interval, then we use the inequality symbols

Calculation:

The inequality: $14\frac{1}{2}<b-1\frac{1}{4}$

  $\begin{array}{l}14\frac{1}{2}<b-1\frac{1}{4}\\ b-1\frac{1}{4}>14\frac{1}{2}[b<c\Rightarrow c>b]\\ b-1\frac{1}{4}+1\frac{1}{4}>14\frac{1}{2}+1\frac{1}{4}\text{[}\text{Add}1\frac{1}{4}\text{to}\text{both}\text{sides}\text{]}\\ b>15\frac{3}{4}\end{array}$

  $\begin{array}{l}\text{Check}\text{the}\text{solution}\text{when}b=20\frac{1}{4}\\ 14\frac{1}{2}<b-1\frac{1}{4}\\ 14\frac{1}{2}<20\frac{1}{4}-1\frac{1}{4}\\ 14\frac{1}{2}<19[\text{True}\text{Statement}]\end{array}$

The solution of $14\frac{1}{2}<b-1\frac{1}{4}$ is $b>15\frac{3}{4}\text{or}(15\frac{3}{4},\infty )$

Thus, the solution of $14\frac{1}{2}<b-1\frac{1}{4}$ is $b>15\frac{3}{4}\text{or}(15\frac{3}{4},\infty )$