2.7 Solve linear inequalities (Page 3/8)

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Properties of equality

\begin{matrix} Division Property of Equality & Multiplication Property of Equality \\ For any numbers a, b, c, and c \neq 0, & For any real numbers a, b, c, \\ \begin{matrix} if & a & = & b, \\ then & \frac{a}{c} & = & \frac{b}{c} . \end{matrix} & \begin{matrix} if & a & = & b, \\ then & a c & = & b c . \end{matrix} \end{matrix}

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.

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.

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{matrix} For any real numbers a, b, c \\ \begin{matrix} if a < b and c > 0, then \frac{a}{c} < \frac{b}{c} and a c < b c . \\ if a > b and c > 0, then \frac{a}{c} > \frac{b}{c} and a c > b c . \\ if a < b and c < 0, then \frac{a}{c} > \frac{b}{c} and a c > b c . \\ if a > b and c < 0, then \frac{a}{c} < \frac{b}{c} and a c < b c . \end{matrix} \end{matrix}

When we divide or multiply an inequality by a:

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

Solve the inequality $7 y < 42$ , 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.

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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$9 c > 72$

$c > 8$
This figure is a number line ranging from 6 to 10 with tick marks for each integer. The inequality c is greater than 8 is graphed on the number line, with an open parenthesis at c equals 8, and a dark line extending to the right of the parenthesis.

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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$12 d \leq 60$

$d \leq 5$
This figure is a number line ranging from 3 to 7 with tick marks for each integer. The inequality d is less than or equal to 5 is graphed on the number line, with an open bracket at d equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.

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Solve the inequality $−10 a \geq 50$ , 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.

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Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$−8 q < 32$

$q > −4$
This figure is a number line ranging from negative 6 to negative 3 with tick marks for each integer. The inequality q is greater than negative 4 is graphed on the number line, with an open parenthesis at q equals negative 4, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 4 comma infinity, parenthesis.

Got questions? Get instant answers now!

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$−7 r \leq - 70$

This figure is a number line ranging from 9 to 13 with tick marks for each integer. The inequality r is greater than or equal to 10 is graphed on the number line, with an open bracket at r equals 10, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 10 comma infinity, parenthesis.

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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{matrix} x > a has the same meaning as a < x \end{matrix}

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

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.

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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$24 \leq \frac{3}{8} m$

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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