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Solving Linear Inequalities in One Variable

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This section marks the beginning of a pattern we'll see repeated regularly throughout the first several chapters: first, we'll talk about a type of equation then we'll talk about the same concept but with the equality changed to an inequality. In this case, a linear inequality is an inequality that can be written in the form

$$ax+b\lt c$$


$$ax+b\le c$$

This also covers the > and ≥

cases since any "greater than" inequality can be rewritten as a "less than" inequality by changing the sides, e.g. $x \gt y$ is equivalent to $y \lt x$.

One of the nice things about linear inequalities is that they can be solved following the same set of steps you previously learned for solving linear equations but with one new rule added.

  1. Multiply both sides to eliminate any fractions or decimals.
  2. Eliminate parentheses.
  3. Simplify each side by combining like terms.
  4. Get all variable terms on one side and all the numbers on the other side.
  5. Divide to get the variable by itself.
    1. If the number you divided (or multiplied) by is negative then change the direction of the inequality.
  6. Check your result.

Video Lectures


Linear inequalities are very similar to linear equations. The procedures for solving them are practically identical but with one extra rule for working with inequalities. (lecture slides)
Just like linear equations, linear inequalities have to special cases you need to be aware of - one where there are no solutions and a second where there are infinitely many solutions. (lecture slides)

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