Matematica

Linear Inequalities: Solved Exercises and the Sign-Flip Rule

How to solve a linear inequality step by step, with the key rule about flipping the inequality sign, plus five solved exercises of increasing difficulty with verified answers.

Recommended for: Grade 8 ยท Grade 9

A linear inequality of the form ax + b > 0 (or with <, >=, <=) is solved like an equation, by isolating x, with one extra rule: if you multiply or divide by a negative number, the inequality sign flips. The solution is not a single number but an interval.

The method, and the one trap

Simplify both sides, move the x-terms to one side and the numbers to the other, then divide by the coefficient of x. The only difference from equations is the sign trap: dividing (or multiplying) by a negative number flips the inequality. It is the most common mistake, and exercise three below is there on purpose to lock it in. Every answer is verified by substituting a value from the interval back into the original inequality.

Solved exercises

1. Solve: 2x + 4 > 0 base

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  1. Move the constant to the right: 2x > -4.
  2. Divide by 2 (positive, sign stays): x > -2.

Answer: x > -2

2. Solve: 3x - 9 <= 0 base

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  1. Move -9 to the right: 3x <= 9.
  2. Divide by 3 (positive): x <= 3.

Answer: x <= 3

3. Solve: 5 - 2x < 1 intermedio

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  1. Move 5 to the right: -2x < -4.
  2. Divide by -2: since it is NEGATIVE, flip the sign -> x > 2.

Answer: x > 2

4. Solve: 4(x - 1) >= 2x + 2 intermedio

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  1. Expand: 4x - 4 >= 2x + 2.
  2. Group terms: 2x >= 6.
  3. Divide by 2: x >= 3.

Answer: x >= 3

5. Solve: (x + 1)/2 - (x - 2)/3 > 1 avanzato

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  1. Multiply everything by 6: 3(x + 1) - 2(x - 2) > 6.
  2. Expand: 3x + 3 - 2x + 4 > 6, so x + 7 > 6.
  3. Move 7 to the right: x > -1.

Answer: x > -1

FAQ

When do you flip a linear inequality sign?

When you multiply or divide both sides by a negative number. The inequality sign reverses (< becomes >, and vice versa).

Is the solution of a linear inequality a single number?

No, it is an interval: infinitely many values. For example x > 2 means every number greater than 2.

How is it different from a linear equation?

The method to isolate x is the same, but an equation has one solution value while an inequality has an interval, and you must remember the sign-flip rule.