Chapter 3.3, Problem 4P

Solve each inequality. Graph and check your solution.

$1\ge -\frac{2}{3}y$

To determine

ToSolve:The inequality $1\ge -\frac{2}{3}y$ . Also, graph and check the solution.

Answer to Problem 4P

The solutions of given inequality are all the real numbers in the interval $\left[\frac{-3}{2},\infty \right)$

The graph of the given inequality is

  

Explanation of Solution

Given:

The inequality $1\ge -\frac{2}{3}y$

Concept Used:

The solution of aninequality is the set of all the values which satisfies the inequality.

The graph of an inequality in one variable shows all the solutions of the inequality on a number line.

Calculation:

Given the inequality $1\ge -\frac{2}{3}y$

Solving it, we have

  $\begin{array}{c}1\ge -\frac{2}{3}y\\ 1\left(\frac{-3}{2}\right)\ge \left(-\frac{2}{3}y\right)\left(\frac{-3}{2}\right)\text{ Multiply by }\frac{-3}{2}\text{ to both sides}\\ \frac{-3}{2}\le y\end{array}$

That is $y\in \left[\frac{-3}{2},\infty \right)$

Therefore, thesolutions of given inequality are all the real numbers in the interval $\left[\frac{-3}{2},\infty \right)$

Checking

At end point $y=\frac{-3}{2}$

Substituting $y=\frac{-3}{2}$ into the given inequality, we have.

  $\begin{array}{c}1\ge -\frac{2}{3}y\\ 1\ge -\frac{2}{3}\left(\frac{-3}{2}\right)\text{ Substitute }y=\frac{-3}{2}\\ 1\ge 1\text{ Which is True}\end{array}$

Substituting a solution from the set of solutions into the given inequality say $y=0$ , we have. $\begin{array}{c}1\ge -\frac{2}{3}y\\ 1\ge -\frac{2}{3}\left(0\right)\text{ Substitute }y=0\\ 1\ge 0\text{ Which is True}\end{array}$

Thus, the solutions of given inequality are all the real numbers in the interval $\left[\frac{-3}{2},\infty \right)$

Graphing the solutions of the given inequality.

Since, the solutions of given inequality are all the real numbers $y$ , such that $y\in \left[\frac{-3}{2},\infty \right)$

Therefore, the graph of the given inequality is

  

Conclusion:

The solutions of given inequality are all the real numbers in the interval $\left[\frac{-3}{2},\infty \right)$

The graph of the given inequality is