Розв'язування нерівностей: властивість ділення

(Video) Solving 1-Step Inequalities

by Friendly Math 101

This video by Friendly Math 101 shows you how to solve one-step inequalities.

$\large{\text{Summary}}$

Solving inequalities is almost the exact same as solving equations. You apply inverse operations to both sides in order to isolate the variable. Instead of having the variable equal one number, with inequalities the variable will be a range of numbers.

Let's look at the last example, $\frac{p}5\ge-3$. In order to undo the division of $5$ we multiply by $5$ on both sides. $$\begin{align} \frac{p}5&\ge-3\\ \frac{p}{5}\color{#29A603}{\times5}&\ge-3\color{#29A603}{\times5}\\ p&\ge-15 \end{align}$$ This means that $x$ can be any number greater than $-15$ including $-15$ itself. We represent this on a number line by drawing a closed circle on the $-15$ which tells us that it can be $-15$. Then an arrow to the right which tells us that it can be values above $-15$.

Dividing by a Positive to Solve an Inequality

Dividing by a positive to solve an inequality is the same as dividing by a positive to solve an equation. Just remember to divide both sides by the same quantity, then simplify! The inequality sign will stay the same the entire time.

Let's do another example!

To solve for $x,$ we simply divide both sides by $30,$ then simplify! Remember, the inequality sign will stay the same.

$$\begin{align} 30x &< 15\\ \dfrac{30x}{30} &< \dfrac{15}{30}\\ \dfrac{\cancel{30}x}{\cancel{30}} &< \dfrac{1}{2}\\ x&<\dfrac{1}{2} \end{align}$$

Solving Inequalities: Division Property

Much like with normal equations, you can use division to reduce an inequality to a simpler form. Specifically, the division property of inequalities is:

When dividing an inequality by a negative number, flip the inequality symbol. When dividing by a positive number, keep the symbol the same.

When $c$ is positive:

Let's say that $c$ is positive and $a$ is greater than $b$. That means that we know that: $$c > 0\\ a > b$$

If we divide $a$ and $b$ by $c$ here, their sign and their relationship stays the same. So, the term with $a$ in it is greater than the term with $b$ in it: $$\frac{a}{c} > \frac{b}{c}$$

$$\begin{align} 4x &> 8\\ \frac{4x}{4} &> \frac{8}{4}\\\ x &> 2\\ \end{align}$$

This makes a lot of sense when you think about it! If you take a big number and a small number and divide them both by the same number, the big one is still going to be bigger than the small number.

The exception here is when you divide by a negative number. Then, the big number will be a "bigger" negative number which means it will actually be less than the smaller negative number.

Multiplying by a Positive to Solve an Inequality

When we multiply by a positive to solve an inequality, it's the same as multiplying by a positive to solve a regular equation! Just remember to multiply both sides by the same quantity, then simplify! The inequality sign will stay the same the entire time.

Let's do another example!

To solve for $x,$ we simply multiply both sides by $8,$ then simplify! Remember, the inequality sign will stay the same.

$$\begin{align} \dfrac{x}{8} &< 4 \\ 8 \cdot \dfrac{x}{8} &< 4 \cdot 8 \\ \cancel{8} \cdot \dfrac{x}{\cancel{8}} &< 32 \\ x&<32 \end{align}$$