Inequalities
A rule says “at least” or “no more than” instead of “equals”. How do you find, draw and write every number that passes?
Where you are 7 inside
What this area is about
A linear equation such as 2x + 1 = 7 is usually true for one number. An inequality such as x > 3 is usually true for a whole stretch of the number line. This area is about finding that stretch, drawing it and naming it, and about the one move that differs from solving equations: multiplying or dividing both sides by a negative number reverses the sign.
How these fit together
Pictures and words. Inequalities on the number line shows a solution set with open or closed circles and shading. Writing an inequality for a condition turns “at most 23 kg” into w \le 23. Interval and set-builder notation writes a stretch compactly, as [3, \infty).
Solving. Solving linear inequalities takes one, two and many steps, with the sign flip.
Two conditions. Compound inequalities join two tests with “and” (the overlap) or “or” (either part). Absolute value inequalities read |w - 340| \le 8 as a distance and become compound inequalities.
Beyond lines. Polynomial and rational inequalities use a sign chart built from zeros and undefined points.
A path through it
The number line first. Then writing a condition, interval notation, and solving linear inequalities; compound inequalities after solving, and absolute value inequalities after compound ones. Polynomial and rational inequalities wait for quadratic inequalities and graphs of rational functions.
After this: systems of inequalities and modeling constraints.