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Linear Inequalities in One and Two Variables

SAT MathAlgebraMedium

Solve inequalities, reverse signs correctly, and interpret shaded regions.

What you need to know

  • Reverse the inequality when multiplying or dividing by a negative number.
  • A strict inequality uses an open boundary; ≤ or ≥ includes the boundary.
  • For two variables, test a point to choose the shaded side.
  • Context may require integer, nonnegative, or bounded solutions.

The AYLOTI method

  1. Solve like an equation until a negative division occurs.
  2. Reverse the sign if required.
  3. Graph the boundary with correct line style.
  4. Test a point and apply contextual restrictions.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

Worked example

Solve −3x + 5 ≥ 17.

  1. Subtract 5: −3x ≥ 12.
  2. Divide by −3 and reverse ≥ to ≤.
  3. x ≤ −4.

Answer: x ≤ −4.

Common SAT traps

  • Forgetting to reverse the sign.
  • Using a solid line for < or >.
  • Including impossible negative counts in context.
Quick check

Does (0, 0) satisfy y > 2x + 1?

No, because 0 > 1 is false.

Before you move on

  • Explain the core rule without looking back.
  • Redo the worked example using your own steps.
  • Write the trap you are most likely to fall for.
Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

Educate A Change focuses on truly bringing change in the educational sector worldwide. All our courses are designed to ensure success, and provide complete preparation without requiring any other external material. Therefore, it takes away the expense of academies, books, past paper compilations and other miscellaneous items. Instead, by just the courses we offer, the student can prepare fully for the best grades in their CAIEs, Edexcel and other high-school exams. The earnings from our course-sales are invested in social projects and growing the free educational activities of Educate A Change.