Why is it important to learn how to solve inequalities? Does it enhance critical thinking? How can it be applied to real life scenarios?
Inequalities are arguably used more often in "real life" than equalities.
Businesses use inequalities to control inventory, plan production lines, produce pricing models, and for shipping/warehousing goods and materials. Look up linear programming or the Simplex method. These methods allow one to maximize profit or minimize cost subject to a wide variety of constraints. (e.g. limited materials, power costs, labor costs, shipping/storing costs, taxes, etc...)
Inequalities are used to limit: you must be taller than 42" to ride a ride at a park, shorter than 6'2" to be a pilot or work in a submarine, weigh at least 100lbs to donate blood, etc...
Inequalities are used in engineering and production quality assurance. When you buy a 2l soda, you are actually buying soda whose volume lies in some range about 2l. Thus producers work within some tolerance which is just a set of inequalities. Engineers work with safety measures and material strengths to make sure structures are safe.
For critical thinking, we are used to working with equalities which are equivalence relations. Inequalities are not equivalence relations (they do not have the symmetric property for example; e.g. a<b does not mean that b<a while a=b <=> b=a.) Working in a different type of system increases your ability to think critically -- you must check your assumptions. (For instance, it seems "obvious" that you can multiply both sides by the same thing and retain a true statement, but 2<3 is true while -2<-3 is false, despite the fact that we multiplied both sides by the same thing.)
All of calculus is based on inequalities. So virtually all of higher math/science uses inequalities extensively. Statistical results are often reported as inequalities. (If a polling result is reported as 47% `+-3%` , what they are reporting is that the true percent is 44<p<50.)
Inequalities are critical in prediction of future results. You know an upper limit, but can't predict where below that upper limit actual results will fall. Using the upper limit as the boundary, and solving the inequality can give you an idea of what may happen, though without certainty.
Inequalities are used constantly in estimation, subjective evaluations, overlapping possibilities, if...then scenarios.
A simple example of inequalities. Al earns $3500/mo at his office job, and he earns up to (an inequality) $1500 per appointment as a securities agent. How many appointments can he go on without going over $5,000/mo?