Chapter 8.7, Problem 42IP

To determine

Solve the inequality and check the solutions.

Answer to Problem 42IP

The solution of $12-d>40$ is $d<-28\text{or}(-\infty ,-28)$

Explanation of Solution

Given:

The Inequality: $12-d>40$

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: $12-d>40$

  $\begin{array}{l}12-d>40\\ 12-12-d>40-12\text{[}\text{Subtract}12\text{from}\text{both}\text{sides}\text{]}\\ -d>28\\ \frac{-d}{-1}<\frac{28}{-1}[\text{Divide}\text{both}\text{sides}\text{by}-1\text{and}\text{reverse}\text{the}\text{inequality}\text{]}\\ d<-28\end{array}$

  $\begin{array}{l}\text{Check}\text{the}\text{solution}\text{when}d=-30\\ 12-d>40\\ 12-(-30)>40\\ 12+30>40\\ 42>40[\text{True}\text{Statement}]\end{array}$

The solution of $12-d>40$ is $d<-28\text{or}(-\infty ,-28)$

Thus, the solution of $12-d>40$ is $d<-28\text{or}(-\infty ,-28)$