When solving an inequality, you must flip the inequality sign when: — Multiplying or dividing by a negative number.Flipping is only required when multiplying or dividing BOTH sides by a negative number.
Solve: x/4 = 3. — x = 12.Multiply both sides by 4: x = 3 × 4 = 12.
Solve the inequality: x + 5 > 9. — x > 4.x > 9 − 5 = 4.
Solve: 4x + 3 = 19. — x = 4.4x = 16 → x = 4.
A system of two linear equations can have how many solutions? — 0, 1, or infinitely many.Three cases: one solution (intersecting lines), no solution (parallel lines), infinitely many (same line).
Solve the system: x = 3 and y = x + 2. What is y? — 5.Substitute x=3 into y=x+2: y=3+2=5.
Two equations with the same slope and different y-intercepts have: — No solution (parallel lines).Same slope = parallel lines. Different y-intercepts means they never intersect — no solution.
Solve by substitution: y = 2x and x + y = 9. — (3, 6).Substitute y=2x: x+2x=9 → 3x=9 → x=3. Then y=2(3)=6. Solution: (3,6).
Which method works best when one equation already has a variable isolated? — Substitution.Substitution is most efficient when a variable is already alone on one side (e.g., y = ...).
Solve by elimination: x + y = 5 and x − y = 1. — (3, 2).Add equations: 2x=6 → x=3. Then 3+y=5 → y=2. Solution: (3,2).
If two equations are identical (same line), the system has: — Infinitely many solutions.If both equations represent the same line, every point on the line is a solution — infinitely many.
What is the standard form of a quadratic equation? — ax² + bx + c = 0.Standard form is ax² + bx + c = 0, where a ≠ 0.
The quadratic formula is: — x = (−b ± √(b²−4ac)) / (2a).x = (−b ± √(b²−4ac)) / (2a). Note the entire numerator over 2a.
What is the discriminant of ax² + bx + c = 0? — b² − 4ac.Discriminant = b² − 4ac. It determines the number and type of roots.
Factor and solve: x² − 5x + 6 = 0. — x = 2 and x = 3.(x−2)(x−3) = 0 → x = 2 or x = 3.
If discriminant = 0, the quadratic has: — One repeated real root.Discriminant = 0 means one repeated root: x = −b/(2a).
Solve: x² = 16. — x = ±4.x² = 16 → x = ±√16 = ±4. Both positive and negative solutions!
If discriminant < 0, the roots are: — Two complex (imaginary) numbers.A negative discriminant means √(negative), which gives complex (imaginary) roots.
The graph of a quadratic function is called a: — Parabola.Every quadratic y = ax² + bx + c graphs as a parabola — a U-shaped curve.
If a > 0 in y = ax² + bx + c, the parabola: — Opens upward.Positive a → parabola opens upward (like a "U"). Negative a → opens downward.
What is the axis of symmetry formula? — x = −b/2a.Axis of symmetry: x = −b/(2a). The parabola is symmetric about this vertical line.
What is the y-intercept of y = 3x² − 2x + 7? — 7.Set x = 0: y = 0 − 0 + 7 = 7. The constant term c is always the y-intercept.
The vertex is the highest point when: — a < 0.When a < 0 (opens downward), the vertex is the maximum point (top of the hill).
For y = x² − 6x + 8, what is the axis of symmetry? — x = 3.x = −b/(2a) = −(−6)/(2·1) = 3.
Where does a parabola cross the x-axis? — At the x-intercepts (zeros/roots).The parabola crosses the x-axis at the zeros, roots, or x-intercepts — where y = 0.
A literal equation contains: — More than one variable (letters).Literal equations have multiple variables (letters), like formulas in science or geometry.
Solve d = rt for r. — r = d/t.Divide both sides by t: r = d/t.
Solve A = lw for w. — w = A/l.Divide both sides by l: w = A/l.
Solve P = 2l + 2w for l. — l = (P − 2w)/2.2l = P − 2w → l = (P−2w)/2.
Solve y = mx + b for x. — x = (y − b)/m.Subtract b: y−b = mx. Divide by m: x = (y−b)/m.
Solve C = 2πr for r. — r = C/(2π).Divide both sides by 2π: r = C/(2π).
What does solving a literal equation for a variable mean? — Isolating that variable on one side.Solving for a variable means isolating it alone on one side, treating all other letters as constants.