Chapter 5.4, Problem 72S

To determine

To solve the inequality:

  $2y+4>2\left(3+y\right)$

And to graph the solution on a number line.

Answer to Problem 72S

No solution.

Explanation of Solution

Given:

  $2y+4>2\left(3+y\right)$

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

  $2y+4>2\left(3+y\right)$

First, simplify it as

  $\begin{array}{c}2y+4>2\left(3+y\right)\\ \Rightarrow 2y+4>6+2y\end{array}$

Now, subtract $2y$ from both sides of the inequality, as

  $\begin{array}{c}2y+4>6+2y\\ \Rightarrow 2y+4-2y>6+2y-2y\\ \Rightarrow 4+0>6+0\\ \Rightarrow 4>6\end{array}$

But, 4 is not greater than 6, thus above is contradiction.

Thus, the given inequality is not true for any value of y, that is the above inequality has no solution.