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Solving Inequalities Using the Division and Multiplication Properties of Inequality

Solving Inequalities Using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

Properties of Equality

\(\begin{array}{cccc}\mathbf{\text{Division Property of Equality}}\hfill & & & \mathbf{\text{Multiplication Property of Equality}}\hfill \\ \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,\text{and}\phantom{\rule{0.2em}{0ex}}c\ne 0,\hfill & & & \text{For any real numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,\hfill \\ \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & a\hfill & =\hfill & b,\hfill \\ \text{then}\hfill & \frac{a}{c}\hfill & =\hfill & \frac{b}{c}.\hfill \end{array}\hfill & & & \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & a\hfill & =\hfill & b,\hfill \\ \text{then}\hfill & ac\hfill & =\hfill & bc.\hfill \end{array}\hfill \end{array}\)

Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

Consider some numerical examples.

. .
Divide both sides by 5. . Multiply both sides by 5. .
Simplify. . .
Fill in the inequality signs. . .
\(\mathbf{\text{The inequality signs stayed the same.}}\)

Does the inequality stay the same when we divide or multiply by a negative number?

. .
Divide both sides by −5. . Multiply both sides by −5. .
Simplify. . .
Fill in the inequality signs. . .
\(\mathbf{\text{The inequality signs reversed their direction.}}\)

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

Here are the Division and Multiplication Properties of Inequality for easy reference.

Division and Multiplication Properties of Inequality

\(\begin{array}{} \text{For any real numbers}\phantom{\rule{0.2em}{0ex}}a,b,c\hfill \\ \begin{array}{c}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a0,\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{c}<\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}acb\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c>0,\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{c}>\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}ac>bc.\hfill \\ \phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}ac>bc.\hfill \\ \phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a>b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c<0,\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{c}<\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}ac

When we divide or multiply an inequality by a:

  • positive number, the inequality stays the same.
  • negative number, the inequality reverses.

Example

Solve the inequality \(7y<\text{​}\text{​}42\), graph the solution on the number line, and write the solution in interval notation.

Solution

.
Divide both sides of the inequality by 7.
Since \(7>0\), the inequality stays the same.		 .
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .

Example

Solve the inequality \(-10a\ge 50\), graph the solution on the number line, and write the solution in interval notation.

Solution

.
Divide both sides of the inequality by −10.
Since \(-10<0\), the inequality reverses.		 .
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .

Solving Inequalities

Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.

\(\begin{array}{ccc}& & x>a\phantom{\rule{0.2em}{0ex}}\text{has the same meaning as}\phantom{\rule{0.2em}{0ex}}a

Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”

Example

Solve the inequality \(-20<\frac{4}{5}u\), graph the solution on the number line, and write the solution in interval notation.

Solution

.
Multiply both sides of the inequality by \(\frac{5}{4}\).
Since \(\frac{5}{4}>0\), the inequality stays the same.		 .
Simplify. .
Rewrite the variable on the left. .
Graph the solution on the number line. .
Write the solution in interval notation. .

Example

Solve the inequality \(\frac{t}{-2}\ge 8\), graph the solution on the number line, and write the solution in interval notation.

Solution

.
Multiply both sides of the inequality by \(-2\).
Since \(-2<0\), the inequality reverses.		 .
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .

This lesson is part of:

Solving Linear Equations II

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