Linear Equations & Inequalities

SAT Math: Introductory Algebra Notes

Linear Equations and Inequalities

  • Algebraic relationships can be represented as equations or inequalities, or graphically.

  • A linear equation maintains the same ratio of change in y to change in x (slope).

  • Slope (m) is defined as:

    • Slope = Change in y / Change in x.

    • Example: If for every 1.0 unit increase in y, x decreases by 2.0, then slope m = -2.

  • The graph of a linear equation is a straight line, and it can also be recognized by the slope which is consistent across the line.

Finding Linear Equations from Given Data

  • Three common methods to write linear equations:

    1. Slope and y-intercept.

    2. Slope and one point on the line (Point-slope formula).

    3. Two points on the line (calculate slope first).

Key Formulas to Memorize

  • Slope-intercept formula:

    • y = mx + b

  • Point-slope formula:

    • y - y1 = m(x - x1)

  • Slope formula for two points:

    • m = (y2 - y1) / (x2 - x1)

Writing the Equation of a Line

  1. Slope and y-intercept:

    • Given: Slope (m = 3) and y-intercept (b = 6)

    • Equation: y = 3x + 6

  2. Slope and a point (e.g., (2, 12)):

    • Substitute into point-slope form:

    • y - 12 = 3(x - 2)

    • Convert to slope-intercept:

    • y = 3x + 6 after distribution and simplification.

  3. Two points (e.g., (1, 9) and (3, 15)):

    • Calculate slope using formula: m = (15 - 9) / (3 - 1) = 3.

    • Use point-slope form:

    • y - 9 = 3(x - 1) -> y = 3x + 6 after simplification.

Graphing Linear Equations

  • Four possible outcomes when graphing:

    • Increasing line (positive slope)

    • Decreasing line (negative slope)

    • Horizontal line (zero slope)

    • Vertical line (undefined slope)

  • To graph:

    • Use y-intercept and slope to plot points.

    • Identify y-intercept (where x = 0) and use slope to determine line direction.

Writing the Equation from Graphs

  • Determine y-intercept (b) and slope (m) from the graph:

    • Use slope formula for any two points to find m.

    • Insert into the equation format: y = mx + b.

Linear Inequalities

  • Definition: Express relationships without equality, using greater than or less than symbols (e.g., >, <).

  • Key terms:

    • Inclusive: with equal to (≥, ≤)

    • Non-inclusive: without equal to (> or <)

Rules of Inequalities

  • Perform the same operations on both sides as with linear equations but switch the inequality direction when multiplying or dividing by a negative number.

  • Example: For x < 4, x can take values less than but not equal to 4.

  • To solve inequalities, cross-check by inserting values back into the inequality.

Graphing Linear Inequalities

  • Solid line for inclusive (≥, ≤) inequalities; dashed line for non-inclusive (> <).

  • Shading regions:

    • Shade below the line for y < (and y ≤).

    • Shade above the line for y > (and y ≥).

  • Use test points (like (0,0)) to determine which side to shade.