College Algebra

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6 < 10

However, if we multiply by –2, we get: –6 > –10

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Solving Linear Inequalities

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Linear Inequalities

An inequality is linear if:

Each term is constant or a multiple of the variable.

To solve a linear inequality, we isolate the variable on one side of the inequality sign.

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E.g. 1—Solving a Linear Inequality

Solve the inequality 3x < 9x + 4 and sketch the solution set.

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E.g. 1—Solving a Linear Inequality

3x < 9x + 4

3x – 9x < 9x + 4 – 9x (Subtract 9x)

–6x < 4 (Simplify)

(–1/6)(–6x) > (–1/6)(4) (Multiply by –1/6 or divide by –6)

x > –2/3 (Simplify)

The solution set consists of all numbers greater than –2/3.

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E.g. 1—Solving a Linear Inequality

In other words, the solution of the inequality is the interval (–2/3, ∞).

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E.g. 2—Solving a Pair of Simultaneous Inequalities

Solve the inequalities 4 ≤ 3x – 2 < 13

The solution set consists of all values of x that satisfy both of the inequalities 4 ≤ 3x – 2 and 3x – 2 < 13

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E.g. 2—Solving a Pair of Simultaneous Inequalities

Using Rules 1 and 3, we see that these inequalities are equivalent:

4 ≤ 3x – 2 < 13

6 ≤ 3x < 15 (Add 2)

2 ≤ x < 5 (Divide by 3)

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E.g. 2—Solving a Pair of Simultaneous Inequalities

Therefore, the solution set is [2, 5)

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Solving Nonlinear Inequalities

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Nonlinear Inequalities

To solve inequalities involving squares and other powers, we use factoring, together with the following principle.

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The Sign of a Product or Quotient

If a product or a quotient has an even number of negative factors, then its value is positive.

If a product or a quotient has an odd number of negative factors, then its value is negative.

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Solving Nonlinear Inequalities