Linear Inequalities

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Definition of Linear Inequalities

Linear inequalities can be described as any statement that have one or two variables whose exponents are one, and where rather than equalities, inequalities are the center of focus. For example, with 3_y_ < 2, the “<” represents less than and the solution set includes all numbers y < 2/3.

So, a Linear Inequality is all about a linear expression with two variables by using any of the relational symbols such as <,>, ≤ or ≥

Important Points on Linear Inequality

  • A linear inequality divides a plane into two parts.
  • The linear inequality would be either ≥ or ≤ when the boundary line is solid.
  • The linear inequality would be either > or <> when the boundary line is dotted.

Simply put, a linear inequality consists of one of the symbols of inequality:

  • < is less than
  • > is greater than
  • ≤ is less than or equal to
  • ≥ is greater than or equal to
  • ≠ is not equal to
  • = is equal to

Just like linear equations, a linear inequality is about the relationship between linear functions. The only difference is the  fact that linear inequalities relate two linear functions. Since these relationships do not have a strict equality, the probable solutions for expressions containing them are a lot more complex than similar expressions that have a strict equality involved. But most of the times, the same rules are used for linear equations as well as linear inequalities, barring a few nuances that need to be considered.

Properties of Linear Inequalities

  1. The inequality stays unchanged in a situation where the same number is added to both sides of the inequality.

For instance: 

(i) x – 2 > 1

⇒ x – 2 + 2 > 1 + 2 (by adding 2 to both sides)

⇒ x > 3

(ii) x < 5

⇒ x + 1 < 5 + 1 (by adding 1 to both sides) 

⇒ x + 1 < 6

(iii) x – 3 > 2

⇒ x – 3 + 3 > 2 + 3 (by adding 3 to both sides) 

⇒ x > 5

  1. The inequality stays unchanged in a situation where the same number is subtracted from both sides of the inequality.

For instance: 

(i) x + 3 ≤ 7

⇒ x + 3 – 3 ≤ 7 – 3 (by subtracting 3 from both sides) 

⇒ x ≤ 4

(ii) x ≥ 4

⇒ x – 3 ≥ 4 – 3 (by subtracting 3 from both sides) 

⇒ x – 3 ≥ 1

(iii) x + 5 ≤ 9

⇒ x + 5 – 5 ≤ 9 – 5 (by subtracting 5 from both sides) 

⇒ x ≤ 4

3. The inequality stays unchanged in a situation where the same positive number is multiplied to both sides of the inequality.

For instance: 

(i) x/3 < 4

⇒ x/3 × 3 < 4 × 3 (Multiplying 3 to both sides.) 

⇒ x < 12

(ii) x/5 < 7

⇒ x/5 × 5 < 7 × 5 (Multiplying 5 to both sides.) 

⇒ x < 35

  1. The inequality stays unchanged in a situation where the same positive number divides both sides of the inequality.

For instance: 

(i) 2x > 8

⇒ 2x/2 > 8/2 (Dividing both sides by 2) 

⇒ x > 4

(ii) 5x > 8

⇒ 5x/5 > 8/5 (Dividing both sides by 5) 

⇒ x > 8/5

6. The inequality would change when the same negative number divides both sides. It reverses.

For instance: 

(i) -3x > 12

⇒ -3x/-3 < 12/-3 (Dividing both sides by -3) 

⇒ x < -4

(ii) -5x ≤ -10

⇒ -5x/-5 ≥ -10/-5 (Dividing both sides by -5) 

⇒ x ≥ 2

(iii) -4x > 20

⇒ (-4x)/(-4) < 20/(-4) (Dividing both sides by -4) 

⇒ x < -5

  1. The inequality would also change in a situation where the same negative number is multiplied to both sides of the inequality. It reverses.

For instance: 

(i) x/5 > 9

⇒ x/5 × (-5) < 9 × (-5)

⇒ -x < -45

⇒ x > 45

(ii) -x > 5

⇒ -x × (-1) < 5 × (-1)

⇒ x < -5

(iii) x/(-2) > 5

⇒ x/(-2) × (-2) < 5 × (-2)

⇒ x < -10

Linear Inequalities of Real Numbers

Two-dimensional linear inequalities are basically expressions that come in two variables of the form:

a + by < c and ax + by ≥ c,

{\displaystyle ax+by<c{\text{ and }}ax+by\geq c,}where the inequalities are either strict or not. When it comes to the solution set of such an inequality, we can graphically represent it by a half-plane (all the points on one “side” of a fixed line) in the Euclidean plane. The line that is responsible for finding out the half-planes (ax + by = c) is not included in the solution set when the inequality is strict. A simple method for calculating which half-plane is in the solution set is to calculate the value of ax + by at a point (x0y0), which is not on the line and then examine whether the inequality is satisfied or not.

For example, for drawing the solution set of x + 3y < 9, we need to first draw the line with equation x + 3y = 9 as a dotted line, which would signify that the line is not included in the solution set because the inequality is strict. After that, we would need to select a convenient point (not on the line), such as (0, 0). Since 0 + 3(0) = 0 < 9. Note that this point is a part of the solution set, so the half-plane consisting of this point (the half-plane “below” the line) is the solution set of this linear inequality.

Applications

Polyhedra

The set of solutions of a real linear inequality is made up of a half-space of the ‘n’-dimensional real space, one of the two defined by the corresponding linear equation.

The set of solutions of a system of linear inequalities always corresponds to the intersection of the half-spaces defined by individual inequalities. Since the half-spaces are convex sets, it is called a convex set, and the intersection of a set of convex sets is also known as convex. But when it comes to non-degenerate cases, this convex set is a convex polyhedron (perhaps boundless, e.g., a half-space, a slab between two parallel half-spaces or a polyhedral cone). It could also be an empty or a convex polyhedron of lower dimension that has been restricted to a subspace of the n-dimensional space Rn.

Linear programming

A linear programming problem is all about optimising (calculating a maximum or minimum value) of a function (known as the objective function) subject to a number of constraints on the variables which, more often than not, are linear inequalities. The list of constraints is a system of linear inequalities.

Practice Problem

Find and graph the solution set of 3x – 4 < 1 – x.

Solution: Firstly, we would need to manipulate the inequality for finding out a corresponding solution set in terms of the independent variable x.

3x – 4 < 1 – x

3x – 4 + 4 < 1 – x + 4

3x < 5 – x

3x + x < 5 – x + x

4x < 5

x < 5/4

We can verify this result by using a value that satisfies x < 5/4; let’s try x= 0.

3(0) – 4 < 1 – (0)

–4 < 1 Inequality holds

Now, it’s time to graph the result. We would specifically need to use an open circle at x = 5/4 as the solution set is a strict inequality (the < symbol is used).

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