Chapter 8.7, Problem 30IP

To determine

Solve the inequality and check the solutions.

Answer to Problem 30IP

The solution of $15\le -\frac{k}{4}-9$ is $k\le -96\text{or}(-\infty ,-96]$

Explanation of Solution

Given:

The Inequality: $15\le -\frac{k}{4}-9$

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: $15\le -\frac{k}{4}-9$

  $\begin{array}{l}15\le -\frac{k}{4}-9\\ 15+9\le -\frac{k}{4}-9+9\text{[}\text{Add}9\text{to}\text{both}\text{sides}\text{]}\\ 24\le -\frac{k}{4}\\ -4·24\ge -4·-\frac{k}{4}[\text{Multiply}\text{both}\text{sides}\text{by}-4\text{and}\text{reverse}\text{the}\text{inequality}\text{]}\\ -96\ge k\\ k\le -96[b\ge c\Rightarrow c\ge \le b]\end{array}$

The solution of $15\le -\frac{k}{4}-9$ is $k\le -96\text{or}(-\infty ,-96]$

Graph on the number line:

A closed, or shaded, circle is used to represent the inequalities greater than or equal to ( $\ge$ ) or less than or equal to ( $\le$ ). The point is part of the solution.

An open circle is used for greater than (>) or less than (<). The point is not part of the solution. The graph then extends endlessly in one direction.

For = and <, the arrow points down to the left.

For = and >, the arrow points up to the right

  

Thus, the solution of $15\le -\frac{k}{4}-9$ is $k\le -96\text{or}(-\infty ,-96]$