Linear Inequalities

Define an inequality and understand its classification

• Two real numbers or two algebraic expressions related by the symbols ‘<’, ‘>’, ‘≤’ or ‘≥’ form an inequality.

  • For example: 6 < 26, 3 < z + 1 ≤ 22, 27 ≥ s ≥ 16, p + t > 100

• Inequalities can be classified as
  

  • Numerical inequality: Inequalities that involve numbers only are classified as numerical inequalities. For example: 87 < 117, 19 > 17 > 8 etc.
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  • Literal inequality: Inequalities that involve a variable on one side and a number on the other side are classified as literal inequalities. For example: a < 6, 18 > k, b ≥ − 27, 21 ≤ m, etc.
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  • Double inequality: Inequalities in which the variable or the numbers lie in a certain interval are known as double inequalities. For example: x ∈ [−15, 8], 9 > 6 > 2, 8 ≤ p + 1 ≤ 11, etc.
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  • Strict inequality: Inequalities of the type px + q < 0, px + q > 0, px + qy < r, px + qy > r, ax² + bx + c > 0, or ax² + bx + c < 0 are classified as strict inequalities. For example: 2x < − 3, x + 17 < 9, x + 3y > 14, 2a + 5b < 8, 2y² + 5y > 8 etc.
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  • Slack inequality: Inequalities of the type px + q ≤ 0, px + q ≥ 0, px + qy ≤ r, px + qy ≥ r, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0 are classified as slack inequalities. For example: x ≤ 89, 5x + 8y ≤ 9, 8x + y ≥ 7, x + 14 ≥ 28, z² + 3z ≤ 30 etc.
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  • Linear inequality in one variable: Inequalities of the type px + q ≥ 0, px + q ≤ 0, px + q > 0, or px + q < 0, where p ≠ 0, are classified as linear inequalities in one variable (here, the variable in each inequality is x). For example: x − 23 ≥ 0, 12y < 85, etc.
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  • Linear inequality in two variables: Inequalities of the type px + qy + r ≥ 0, px + qy + r ≤ 0, px + qy + r > 0, or px + qy + r < 0, where p ≠ 0 and q ≠ 0, are classified as linear inequalities in two variables (here, the variables in each inequality are x and y). For example: 9x + y > 0, x + 11y ≥ 13, etc.
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  • Quadratic inequality: Inequalities of the type ax² + bx + c ≥ 0, ax² + bx + c ≤ 0, ax² + bx + c > 0, or ax² + bx + c < 0, where a ≠ 0, are classified as quadratic inequalities. For example: x² + 16 ≥ 23, p² < 2p + 7, etc.

Let's now try and solve the following puzzle to check whether we have understood this concept.

Let us now solve an example based on inequality.

Example 1:

State true or false for the following statements:

1. The inequality 9x² + 5x < 0 is a quadratic inequality.
2. The inequality 7p + 3q > 2 is a linear inequality in one variable and a slack inequality.
3. The inequality 8 ≥ p + q ≥ 2 is a double linear inequality in two variables.
4. The inequality 2k + 1 ≤ 8 is a numerical inequality.
5. The inequality s > 100 is not a strict inequality.

Solution:

1. True.
2. False. The inequality 7p + 3q > 2 is a linear inequality in two variables.
3. True.
4. False. The inequalit…