Chapter 5.4, Problem 3BCYP

To determine

To solve the compound inequality:

  $x\le 9,\text{ or }2+4x<10$

And to graph the solution on a number line.

Answer to Problem 3BCYP

  $\left(-\infty ,2\right)$

  

Explanation of Solution

Given:

  $x\le 9,\text{ or }2+4x<10$

Concept Used:

• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.endpoint.
• To graph endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Properties of inequality:
1. $b\le c\Rightarrow b\pm a\le c\pm a$
2. $b\le c\Rightarrow ab\le ac,\text{ if }a>0$
3. $b\le c\Rightarrow ab\ge ac,\text{ if }a<0$

Calculation:

In order to solve the compound inequality:

  $x\le 9,\text{ or }2+4x<10$

First, solve the inequality

  $2+4x<10$

For that first subtract 2 from both sides, as

  $\begin{array}{c}2+4x-2<10-2\\ \Rightarrow 4x<8\\ \Rightarrow \frac{4x}{4}<\frac{8}{4}\\ \Rightarrow x<2\end{array}$

Combining above:

  $x\le 9,\text{ and }x<2\Rightarrow x<2$

Thus, the solution of given inequality is every number strictly less than 2, this can be represented in interval form, as

  $\left(-\infty ,2\right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point 2 is strict, so use a hollow circle at 2. And the solution interval has $-\infty$ at left endpoint which means every number strictly less than 2 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: