Inequalities

The word inequality means a mathematical expression in which the sides are not equal to each other. Basically, an inequality compares any two values and tells something about their relative sizes.

There are five inequality symbols used to represent equations of inequality.

In general we treat inequalities just like equalities except if we multiply or divide both sides by the same negative number we reverse the direction of the inequality symbol.

We generally like to write our answers with the variable written on the left hand side of the inequality symbol. If it happens to be the other way round just change the order and reverse the inequality symbol. That is effectively multiply both sides by -1.

Example

Calculate the range of values of y, which satisfies the inequality: y − 4 < 2y + 5.

Solution

Add both sides of the inequality by 4.

y – 4 + 4 < 2y + 5 + 4

y < 2y + 9

Subtract both sides by 2y.

y – 2y < 2y – 2y + 9

-Y < 9

Multiply both sides of the inequality by −1 and change the inequality symbol’s direction.

y > − 9

Solve the inequality |x-3| > 5
      (1) by algebra
      (2) by sketching a graph

(1) If we remove the modulus (absolute value) signs, x-3 may be positive or it may be negative.
We need to solve for each possibility.

If x-3 is negative: x-3 < -5
x < -5 + 3
x < -2

If x-3 is positive: x-3 > 5
x > 5 + 3
x > 8

Combining both gives Solution: -2 > x and x > 8

(2) Graphs of the modulus of linear functions are “V” shaped.

The vertex of the V will be on the x-axis where the value of x makes the function equal to
zero. In this case at x = 3.
Draw a line given by y = the number the function is compared to. In this case y = 5.
Pick a value for x to the right of the vertex and compute the value of the function so that you
have another point that allows you draw the blue line going up to the right. The blue line
going up to the left will be a reflection of this in a vertical line through the vertex.

You can now read off the graph what values of x cause the graph to be under, on or above the
red line.

Don’t multiply or divide by a variable (unless you know it is always positive or always negative).