Chapter 5.3, Problem 11CYU

To determine

To solve the inequality:

  $3-8x\ge 9+2\left(1-4x\right)$

And to graph the solution on a number line.

Answer to Problem 11CYU

No solution.

Explanation of Solution

Given:

  $3-8x\ge 9+2\left(1-4x\right)$

Concept Used:

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

Calculation:

In order to solve the inequality

  $3-8x\ge 9+2\left(1-4x\right)$

First, simplify this inequality as, $\begin{array}{c}3-8x\ge 9+2\left(1-4x\right)\\ \Rightarrow 3-8x\ge 9+2-8x\\ \Rightarrow 3-8x\ge 11-8x\end{array}$

Now, adding $8x$ both sides of the inequality gives,

  $\begin{array}{c}3-8x\ge 9+2-8x\\ \Rightarrow 3-8x+8x\ge 11-8x+8x\\ \Rightarrow 3\ge 11\end{array}$

But 3 is neither greater than 11 nor equals to 11, thus above is a contradiction.

Thus, the given inequality doesn’t have a solution.