Linear or first degree inequalities . They are inequalities involving one or more unknowns , numbers and one of the inequality signs (“>”, “<“, “≥”, “≤”), which are verified for certain values of the unknowns. These inequalities and systems of them have much use in linear program problems.
Summary
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- 1 Examples
- 2 Resolution
- 1 Examples
- 3 Representation of the solution
- 1 Example
- 4 Geometric sense of the first degree inequalities
- 1 Proposition
- 2 Examples
- 5 See also
- 6 Sources and references
Examples
- The solution of the linear inequality x + 1> 0 is x> -1.
- The solution of the linear inequality x + 2> 2x is x <2.
Resolution
To solve a linear inequality with an unknown variable, the values of the latter must be found for which the inequality is satisfied. The solution of an inequality is an interval. To find it, the polynomial expression must be simplified in the same way as in first-degree equations, but when dividing the inequality by a negative number, the sign of the inequality must be changed.
Examples
- To obtain the solution of the inequality -2x <4 divide the inequality by the negative number -2, obtaining x> -2.
- To solve the inequality -3x +5> 5x -3 monomials with a literal part are isolated on one side of the sign: -3x -5x> -3 -5, to add the monomials: -8x> -8, and obtain the interval: x <1.
Representation of the solution
The solution of an inequality can be represented on the real line, indicating the extremes of the interval. If one end is included in the interval (when the signs are “greater than or equal to” or “less than or equal to), it is indicated by an opaque point on the line. If the end is not included, it is indicated with an empty dot.
Example
- The representation of the solution x> 7 is
Values greater than 7 are represented to the right of the number line and do not include 7.
Geometric sense of first degree inequalities
First-degree equations are known to represent planes, lines, and points. What does a first degree inequality, or a system of first degree inequalities, represent on the plane?
Two points C and D that are not in a plane (on a given line, in the case of the plane) are said to be on different sides from the plane (line) if segment CD intersects the plane (line). If they are on the same side, CD does not cut the plane (the line).
Proposition
Points C and D are on the same side with respect to the (straight) plane L = 0 if only if the results L (C) with L (D) of the replacement of the coordinates of these points in the expression L give the same sign . [one]
Examples
- Let the equation L = 2x – 3y -5 = 0 be the equation of a line; Points C (3; 4) and D (5; 6) are given. From the substitutions L (C) = 2 (3) -3 (4) -5 <0 and L (D) = 2 (5) – 3 (6) – 5 <0, it is seen that they have the same sign, therefore they are on the same semi-plane, on the same side of line L; precisely in the semi-plane that contains the (0, 0).
- The equation of the plane P = x + 2y + 3z + 5 = 0 is given in the space R 3. Given the points M (1, 0, 1) and N (0, -5,1) let’s see how these points are with respect to the plane P. Replacing: P (M) = 1 + 2 (0) + 3 (1) + 5> 0; P (N) = 1 (0) + 2 (-5) + 3 (1) + 5 <0 as the points have different sign, M and N are in different half-spaces with respect to the P plane and the segment MN cuts to the P plane .
