GED and HiSET mathematics — systems of equations and inequalities

Two kinds of algebra question come up again and again on the GED Mathematical Reasoning test and the HiSET Mathematics test: a pair of equations that must both be true at once, and an inequality that describes a limit, such as a budget, a weight capacity or a minimum number of hours. Both are easy to set up badly and easy to finish badly. A system solved by elimination only works if the right terms cancel; an inequality divided by a negative number is wrong unless the sign is turned around. This material works through both, with adult situations.

The twelve quiz questions start with systems. You solve one system by elimination and one by substitution, read the crossing point of two graphed lines as the solution of the system, and pick the system with no solution by spotting parallel lines with equal slopes and different intercepts. Two word problems turn a situation into a system: tickets sold at a community theater, where the number of tickets and the money taken in give two equations, and two phone plans, where you find the number of texts at which they cost the same. The inequality questions ask you to solve $-3x + 7 > 22$, where the sign must flip, to choose the number-line graph of $x\le -2$ by its circle and arrow, to solve a compound inequality, to write the inequality for an elevator's weight limit, to find the largest whole number of boxes that limit allows, and to test whether an ordered pair lies in the region of a two-variable inequality.

The wrong options come from the errors people actually make: forgetting to flip the sign, an open circle where the endpoint is included, dividing by the wrong number when two plans are compared, giving the number of child tickets when adult tickets were asked, or using a strict limit where "no more than" allows equality. Each explanation shows where the error enters.

The flashcards review elimination and substitution, what a solution of a system is, systems with no solution or infinitely many, inequality symbols, when to flip the sign, open and closed circles, and compound inequalities.

The written work asks for full solutions on paper: a system by elimination, a catering order with two meal prices, choosing between two truck-rental plans, explaining a system with no solution, solving and graphing inequalities, and budget and earnings limits. The oral exam asks one question at a time and expects you to explain each step, especially every time an inequality sign changes direction.

The material is independent practice written by Zestly, based on the published content of the GED and HiSET mathematics tests; it is not produced or endorsed by GED Testing Service or ETS.

• Solve systems of two linear equations by elimination and by substitution
• Interpret the solution of a system as the intersection of two lines and recognize systems with no solution
• Set up and solve a system from a word problem
• Solve linear inequalities, reversing the sign when multiplying or dividing by a negative number
• Graph solution sets on a number line with open and closed circles and solve compound inequalities
• Write and solve inequalities for budgets, limits and minimums, and test points in two-variable inequalities

Practice material written by Zestly, based on the published content of the GED Mathematical Reasoning test (algebraic problem solving: systems of linear equations, linear inequalities and their graphs) and the HiSET Mathematics test (algebraic concepts).