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Solving Linear Inequalities

We know that a linear equation is an equation of the form ax + b = 0. If we replace the equality symbol in a linear equation with an inequality symbol, we have a linear inequality.

 

Linear Inequality

A linear inequality in one variable x is any inequality of the form ax + b < 0, where a and b are real numbers, with a ≠ 0. In place of < we may use ≤, >, or ≥.

 

Inequalities that can be rewritten in the form of a linear inequality are also called linear inequalities.

Before we solve linear inequalities, let's examine the results of performing various operations on each side of an inequality. If we start with the inequality 2 < 6 and add 2 to each side, we get the true statement 4 < 8. Examine the results in the following table.

Perform these operations on each side of 2 < 6:

  Add 2 Subtract 2 Multiply by 2 Divide by 2
Resultin inequality 4 < 8 0 < 4 4 < 12 1 < 3

All of the resulting inequalities are correct. However, if we perform operations on each side of 2 < 6 using -2, the situation is not as simple. For example, -2 · 2 = -4 and -2 · 6 = -12, but -4  is greater than -12. To get a correct inequality when each side is multiplied or divided by -2, we must reverse the inequality symbol, as shown in the following table.

Perform these operations on each side of 2 < 6:

  Add -2 Subtract -2 Multiply by -2 Divide by -2
Resulting inequality 0 < 4 4 < 8 -4 > -12 -1 > -3
     

 
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