Equations & Inequalities formulas

All 20 formulas in this topic

Linear Equationax+b=0  ⟹  x=−baax + b = 0 \implies x = -\frac{b}{a}Solve for the value of x in any linear equation by rearranging to isolate x, finding the unique solution where the expression equals zero.Two-Step Equationsax+b=c  ⟹  x=c−baax + b = c \implies x = \frac{c - b}{a}Solve two-step equations of the form ax + b = c by undoing operations in reverse order to isolate the variable using inverse operations.Literal EquationsA=ℓw  ⟹  w=AℓA = \ell w \implies w = \frac{A}{\ell}Literal equations rearrange formulas to solve for any variable, letting you find missing measurements when you know the others.Solving by Substitution{y=mx+ba2x+b2y=c2\begin{cases} y = mx + b \\ a_2x + b_2y = c_2 \end{cases}Solving by Substitution solves a system of equations by replacing one variable with its equivalent expression, turning two equations into one.Solving by Elimination{a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}Solve a system of two linear equations by adding or subtracting to eliminate one variable, then solve for the remaining variable.Systems of Three Equations{a1x+b1y+c1z=d1a2x+b2y+c2z=d2a3x+b3y+c3z=d3\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}Systems of three equations let you find three unknowns from three relationships, the standard approach when a real-world situation has multiple constraints.Linear Inequalitiesax+b<c,a<0  ⟹  flip the signax + b < c, \quad a < 0 \implies \text{flip the sign}Solve linear inequalities by isolating the variable, flipping the inequality sign when multiplying or dividing by a negative number.Compound Inequalitiesa<x<ba < x < bCompound inequalities describe a range where a variable falls between two values; use them when bounds constrain something from both above and below.Absolute Value Inequalities∣x∣<a  ⟺  −a<x<a|x| < a \iff -a < x < aAbsolute value inequalities express ranges of values at a fixed distance from a point. Use them to solve distance, tolerance, or deviation problems.Solving Quadratics by Factoring(x−r1)(x−r2)=0  ⟹  x=r1,r2(x - r_1)(x - r_2) = 0 \implies x = r_1, r_2Solve quadratic equations by factoring the polynomial into binomials, then apply the Zero Product Property to find the solutions.Zero Product Propertyab=0  ⟺  a=0 or b=0ab = 0 \iff a = 0 \text{ or } b = 0A product equals zero if and only if at least one factor equals zero; use this property to solve equations by factoring.Rational EquationsP(x)Q(x)=0  ⟹  P(x)=0,;Q(x)≠0\frac{P(x)}{Q(x)} = 0 \implies P(x) = 0, ; Q(x) \neq 0A rational equation has a polynomial fraction equal to zero; solve by setting the numerator to zero, then verify the denominator is nonzero.Radical Equationsf(x)=g(x)  ⟹  f(x)=g(x)2\sqrt{f(x)} = g(x) \implies f(x) = g(x)^2Solve equations with square roots by squaring both sides to eliminate the radical, then check for extraneous solutions that don't satisfy the original equation.Exponential Equationsax=b  ⟹  x=log⁡aba^x = b \implies x = \log_a bSolve exponential equations where an unknown exponent needs to be found using logarithms to transform them into solvable linear form.Logarithmic Equationslog⁡ax=b  ⟹  x=ab\log_a x = b \implies x = a^bLearn to solve logarithmic equations by converting to exponential form, the key skill for finding unknown values hidden inside logarithms.Distance Rate Timed=rtd = rtCalculate distance, rate, or time in any uniform motion problem using the formula d = rt, where distance equals rate multiplied by time.Work Rate Problems1t1+1t2=1T\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{T}Work Rate Problems help you find how long a job takes when people work together, using the sum of individual work rates.Mixture Problemsc1v1+c2v2=cf(v1+v2)c_1 v_1 + c_2 v_2 = c_f (v_1 + v_2)Mixture Problems help you find how much of each different-strength solution to combine to achieve a target concentration.Consecutive Integer Problemsn,;n+1,;n+2,…n, ; n+1, ; n+2, \ldotsSet up equations where unknowns represent consecutive integers to find specific number sequences meeting given conditions.Extraneous Solutionsx=s solves the squared equation but not the originalx = s \text{ solves the squared equation but not the original}A solution to the transformed equation that doesn't satisfy the original; found when squaring both sides or applying other non-reversible operations.

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