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Solving Linear Inequalities

These examples illustrate the properties that we use for solving inequalities.

 

Properties of Inequality

Addition Property of Inequality

If the same number is added to both sides of an inequality, then the solution set to the inequality is unchanged.

Multiplication Property of Inequality

If both sides of an inequality are multiplied by the same positive number, then the solution set to the inequality is unchanged.

If both sides of an inequality are multiplied by the same negative number and the inequality symbol is reversed, then the solution set to the inequality is unchanged.

 

Because subtraction is defined in terms of addition, the addition property of inequality also allows us to subtract the same number from both sides. Because division is defined in terms of multiplication, the multiplication property of inequality also allows us to divide both sides by the same nonzero number as long as we reverse the inequality symbol when dividing by a negative number.

Equivalent inequalities are inequalities with the same solution set. We find the solution to a linear inequality by using the properties to convert it into an equivalent inequality with an obvious solution set, just as we do when solving equations.

 

Example 1

Solving inequalities

Solve each inequality. State and graph the solution set.

a) 2x - 7 < -1

b) 5 - 3x < 11

Solution

a) We proceed exactly as we do when solving equations:

2x - 7 < -1 Original inequality
2x < 6 Add 7 to each side
x < 3 Divide each side by 2.

The solution set is written in set notation as {x | x < 3} and in interval notation as (-∞, 3). The graph is shown below:

b) We divide by a negative number to solve this inequality.

5 - 3x < 11 Original equation
-3x < 6 Subtract 5 from each side
x > -2 Divide each side by -3 and reverse the inequality symbol

The solution set is written in set notation as {x | x > -2} and in interval notation as (-2, ∞). The graph is shown below:

 

Example 2

Solving inequalities

Solve State and graph the solution set.

≥ -4 Original inequality
≤ -5(-4) Multiply each side by -5 and reverse the inequality symbol.
8 + 3x ≤ 20 Simplify
3x ≤ 12 Subtract 8 from each side.
x ≤ 4 Divide each side by 3.

The solution set is (-∞, 4], and its graph is shown below:

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