Chapter 3.2, Problem 37PPE

To determine

To find:

Solution of the inequality $\frac{1}{4}+a\ge -\frac{3}{4}$ . To graph it and to check its solutions.

Answer to Problem 37PPE

Solution of the given inequality in interval form is $a=[-1,\infty )$ that is shown red in below graph:

  

Explanation of Solution

Given information:

Inequality $\frac{1}{4}+a\ge -\frac{3}{4}$ is given.

Concept used:

Get the solution of given inequality by removing all terms, associated with unknown variable. Do just opposite operation of that term, each side to eliminate its term.

Calculation:

Subtract $\frac{1}{4}$ each side in given inequality,

  $\begin{array}{l}\frac{1}{4}-\frac{1}{4}+a\ge -\frac{3}{4}-\frac{1}{4}\\ \Rightarrow a\ge -\frac{4}{4}\\ \Rightarrow a\ge -1\end{array}$

On number line, show this solution as

Check: Substitute a value from above solution set, say, $a=0$ in given inequality,

  $\begin{array}{l}\frac{1}{4}+0\ge -\frac{3}{4}\\ \Rightarrow \frac{1}{4}\ge -\frac{3}{4}\end{array}$

that is true, so given solution set is correct.

Conclusion:

So, solution of the given inequality is the set of all the real values, those are greater than or equal to $-1$ , as shown in above graph also.