Solving Linear Inequalities in One Variable

(vertically - using the balance beam )

Solve Equation with a Balance Beam

Pattern: ax + b ≥ cx + d

Both sides simplified (a ,b, c, d are integers.)

Look at the coefficients of x and determine which is the larger integer (furthest to the right on the number line). If c > a then we will keep the variiablle x on that siide of the equation and keep the constant on the other side. To do this we first add opposites on the balance beam below the equation. Look at the pattern, and then follow the same steps through several examples

Solve simplified equations vertically - using the balance beam.

Pattern: c > a

1) Add opps:		c > a → c - a > 0
Complete the step:	(b - d) ≥ (c - a)x →	Let A = (c - a) and B = (b - d) A, B are integers, A > 0
2) Multiply recip: Then:	→ →	Since and A > 0 is coefficient of x

Equivalent Property

This means that all replacement values for x will be on or to the left of

NOTE: Since A > 0 all signs in its path remain the same. This is the advantage of choosing the side of larger coefficient when simplifying the problem.