Factor out common factors throughout the inequalities you are working with to express it in its simplest form. For example, considering the following two inequalities: 6x + 3 > 12y and 4y -- 8 ≥ 12x. Simplify them by dividing through the two inequalities by three and four, respectively, which results in 3x +1 > 4y and y -- 2 ≥ 3x, in that order.
Change the inequality symbol to its corresponding opposite symbol, such as > to <, in the event that during simplifying you have to divide or multiply through the inequality using negative numbers. For example, using the inequality y -- 2 ≥ 3x, if you divide through by -1, the inequality ends up as -- y + 2 ≤ - 3x, which you can best rearrange as 2 ≤ y -- 3x.
Derive linear equations from the linear inequalities by replacing the inequality symbols with an equal sign. For example, from 3x + 1 > 4y you get 3x + 1 = 4y. Construct lines (boundaries) on a Cartesian plane using the derived linear equations and mark along them with the same linear equation. Make a dashed line if the linear inequality for the line had either the symbol > or <, and a continuous line for lines from linear inequalities with either ≤ or ≥ as their inequality symbols.
Pick out points on either side of the lines you end up with on the Cartesian plane, replace the coordinates in the respective inequality, and check whether or not the point satisfies the linear inequality. For example, for y -- 2 ≥ 3x, the point (4,4) does not satisfy the inequality since 2 is not greater than or equal to 12 after replacing the coordinates in the inequality. Shade on the Cartesian plane the side of the line that does not satisfy the inequality.