অসমতা সমাধান: ঋণাত্মক সংখ্যা দিয়ে গুণ

Solving Inequalities: Multiplication by a Negative Number:

Sometimes, it is necessary to multiply an inequality by a negative number in order to solve the inequality. Compared to multiplying inequalities by positive numbers, there's one extra step. It's called flipping the inequality sign. Let's see what exactly this looks like.

Take the following example: $$-x + 4 \leq -8$$

The $-x$ is on the left-hand side. We'll try to move everything else to the other side.

We can use our inverse property of addition (subtracting $4$ from both sides). We can reduce this to: $$-x \leq -12$$

We could add to both sides in order to solve this: $$\begin{align} -x &\leq -12\ -x + 12 &\leq - 12 + 12\

• x + 12 &\leq 0 \
  \ -x + 12 &\leq 0\ -x + 12 + x &\leq 0 + x\ 12 &\leq x\
  \ x &\geq 12 \end{align}$$

But there's a faster way. Much like with equations, we can just multiply both sides by $-1$. $$\begin{align} -x &\leq -12\\ -1(-x) &\geq -1(-12)\\ x &\geq 12 \end{align}$$

In words, the variable $x$ is greater than or equal to $12$.

Careful!

Notice how the sign of the inequality flipped when we multiplied by negative one. Much like when you divide by a negative number, the sign of the inequality must flip! Here's why:

Imagine comparing the numbers $5$ and $3$. Clearly $5 > 3$. But what about $5(-1) > 3(-1)$? The result is $-5 > -3$?!?! Obviously this is wrong! When you multiply both sides by a negative value you make the side that is greater have a "bigger" negative number, which actually means it is now less than the other side! This is why you must flip the sign whenever you multiply by a negative number.

You can also consider this with a number line. On a number line, $5$ is to the right of $3$. However, $-5$ is to the left of $-3$.

As an additional note—much like with division, do not multiply both sides of an inequality by a variable unless you know the sign of the variable!

Okay, now click over to the problems tab and give this technique a try!

$$\textbf{Multiplying and Dividing by a Negative to Solve an Inequality}$$

When we multiply or divide by a negative to solve an inequality, it's the same as multiplying or dividing by a negative to solve a regular equation! Just remember to multiply or divide both sides by the same quantity, then simplify! Remember, when multiplying or dividing by a negative YOU MUST FLIP THE INEQUALITY SIGN.

$$\textbf{Multiplying by a Negative}$$

Let's do an example!

To solve for $x,$ we simply multiply both sides by $-15,$ then simplify! Remember, you have to switch the inequality sign!.

$$\begin{align} \dfrac{x}{-15} &\leq -2 \\ \\ -15 \cdot \dfrac{x}{-15} &\leq -2 \cdot -15 \\ \\ \cancel{-15} \cdot \dfrac{x}{\cancel{-15}} &\leq 30 \\ \\ x &\leq 30\\ \\ x &\geq 30, \text{ since we have to change the sign!} \end{align}$$

$$\textbf{Dividing by a Negative}$$

Let's do an example!

To solve for $x,$ we simply divide both sides by $-21,$ then simplify! Remember, you have to switch the inequality sign!

$$\begin{align} -21x &< 7\\ \\ \dfrac{-21x}{-21} &< \dfrac{7}{-21}\\ \\ \dfrac{\cancel{-21}x}{\cancel{-21}} &< \dfrac{-1}{3}\\ \\ x&<\dfrac{-1}{3}\\ \\ x&>\dfrac{-1}{3}, \text{ since we have to switch the sign!} \end{align}$$

(Video) Solving Basic Linear Inequalities Involving Multiplication or Division

by PatrickJMT

Here Patrick JMT has a good video explaining how to go about solving a few inequalities using division.

$\large{\text{Summary}}$

We can multiply and divide to solve inequalities just like we do with equations. But, there is one catch.

Whenever we multiply or divide by a negative number, we have to flip the inequality sign.

You'll need to keep track of whether you're dividing by a positive or negative number.

Suppose we have the inequality: $$-2x<4$$ To solve this, we need to divide both sides by $-2$. Since we're dividing by a negative, the inequality sign flips: $<$ becomes $>$. $$\begin{align} -2x&<4\\ \frac{-2x}{\color{#29A603}{-2}}&<\frac{4}{\color{#29A603}{-2}}\\ x&>-2 \end{align}$$