Question

How do you graph the solution of $\dfrac{2}{3}\left( {12 - x} \right) > 4$ on a number line?

Answer 1

Hint: We have given the inequality in one variable. There are two types of inequality: Slack inequality and strict inequality. In slack inequality $ \leqslant $ or $ \geqslant $ sign is used and in strict inequality $ < $ or $ > $ sign is used. To simplify the inequality, we perform the same operation as we did to solve a linear equation in one variable.

Complete step-by-step solution:
Step1: We have given the inequality $\dfrac{2}{3}\left( {12 - x} \right) > 4$. To draw the graph of the solution on a number line, first, we simplify the above inequality.
First, multiply by $3$ on both side of the above inequality, we get
$
  3 \times \dfrac{2}{3}\left( {12 - x} \right) > 3 \times 4 \\
   \Rightarrow 2\left( {12 - x} \right) > 12 \\
 $
Step2: Now we divide by $2$ on both side of the above inequality, w egt
 $
   \Rightarrow \dfrac{{2\left( {12 - x} \right)}}{2} > \dfrac{{12}}{2} \\
   \Rightarrow 12 - x > 6 \\
 $
Step3: Now we subtract $12$ from both side, we get
$
   \Rightarrow 12 - x - 12 > 6 - 12 \\
   \Rightarrow - x > - 6 \\
 $
Step 4: Now at last, we multiply the above inequality by $ - 1$ . Here, we have to remember that by multiplying an inequality by a negative sign, the sign of inequality gets changed. So now we get
$
   \Rightarrow - x \times - 1 > - 6 \times - 1 \\
   \Rightarrow x < 6 \\
 $
This means that all the values of $x$ , which are less than $6$ lie on the number line are the part of the solution. Here, remember that $6$ is not included in the solution (or not the part of the solution).
Step 5: Now we draw the graph. For that, we shade all the values less than $6$ . As $6$ is not part of the solution so we draw an open circle at $6$ , which means that $6$ is not included.
Following is the graph of the above inequality

Note: While drawing the graph of slack inequality number is included in the solution, while in strict inequality number is not included.
When we multiply an inequality by a negative number, the sign of inequality gets changed.