PSAT Algebra — inequalities in two variables and shaded regions

The PSAT/NMSQT lists linear inequalities in one or two variables among its Algebra skills, and the two-variable case is where many students lose points: a point that looks fine in one inequality fails the other, a dashed line is read as solid, or "at most" is turned into the wrong sign. This material concentrates on inequalities in two variables and on systems of two of them.

You test ordered pairs against an inequality and against a system, paying attention to points that sit exactly on a boundary line: they belong to the solution set of an inequality with ≤ or ≥ and are excluded by < or >. You find a parameter that makes a point a solution, and you pick out the one point that separates a non-strict inequality from its strict version.

Two questions show the solution set as a shaded region on a coordinate grid. You read the boundary line from its marked points, decide the direction of the sign from the side that is shaded, and decide between strict and non-strict from the line style. For a system you do this for each line and check a point inside the region.

The context questions use adult situations: a caterer's budget for sandwiches and salads, a theater group's minimum ticket revenue, a delivery van limited by both the number of boxes and their weight, and a person dividing a work week between tutoring and lifeguarding. You write the inequality or the system from the words, interpret its constants, and use it to decide whether a plan works or what the least number of hours is.

The quiz has twelve multiple-choice questions with explanations that check every option. The flashcards collect the translation rules (at most, at least), the meaning of solid and dashed boundaries and the test-point method. The printable written work asks you to test points, graph an inequality in words, write systems from situations and justify each step, in the style of a student-produced response. The oral exam gives you one situation at a time and asks you to explain your reasoning to an examiner.

The content is based on the College Board's published skill descriptions for the PSAT/NMSQT Math section (Algebra: linear inequalities in one or two variables). It is independent practice, not produced or endorsed by the College Board.

  • Decide whether a point satisfies a linear inequality in two variables or a system of two inequalities
  • Tell solid from dashed boundaries and include or exclude boundary points accordingly
  • Write a linear inequality or a system of inequalities from a budget, revenue, weight or time constraint
  • Interpret the constants and coefficients of an inequality in context
  • Match a shaded region on a coordinate grid with its inequality or system

Practice material written by Zestly, based on the College Board's skill descriptions for the digital PSAT/NMSQT Math section (Algebra: linear inequalities in one or two variables — creating, interpreting and graphing inequalities and systems of inequalities; interpreting a point in the xy-plane in terms of the solution set), as published on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.

Sample question

A coordinate grid with a dashed line through the marked points (0, 1) and (4, 3); the region above the dashed line is shaded.

The graph shows a dashed line through the marked points $(0, 1)$ and $(4, 3)$, with the region over the line shaded. Which inequality is represented by the graph?

See the answer

$y > \frac{1}{2}x + 1$

The line has slope $\frac{3 - 1}{4 - 0} = \frac{1}{2}$ and $y$-intercept $1$, so it is $y = \frac{1}{2}x + 1$. The shading above the line means $y$ is greater than $\frac{1}{2}x + 1$, and the dashed line means points on it are not included, so the inequality is strict: $y > \frac{1}{2}x + 1$. A solid line would call for $\ge$; shading below would call for $<$.

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