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Algebra Inequality Solvers


Free step-by-step solvers for six families of algebraic inequalities. Each tool walks through the full reasoning, handles sign-chart logic, and reports the final answer in interval notation. Pick a family below to open its solver.
Linear
Linear Inequality Solver
ax+b  <  >  ≤  ≥  cx+dax + b \;{<}\;{>}\;{\leq}\;{\geq}\; cx + d
A linear inequality is the inequality version of a linear equation — same first-power xx, but with <<, >>, ≤\leq, or ≥\geq in place of ==. The solver isolates xx exactly the way it would for a linear equation, with one extra rule: multiplying or dividing both sides by a negative number flips the inequality direction, and the solver tracks every flip explicitly. Once xx is isolated, the answer is reported in interval notation, with parentheses for strict inequalities (<<, >>) and brackets for non-strict (≤\leq, ≥\geq). Identities (every real number satisfies it, e.g. 0<10 < 1) and contradictions (no real number satisfies it, e.g. 0>10 > 1) are handled as named cases.
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Quadratic
Quadratic Inequality Solver
ax2+bx+c  <  >  ≤  ≥  0ax^2 + bx + c \;{<}\;{>}\;{\leq}\;{\geq}\; 0
A quadratic inequality asks where a parabola sits above or below the xx-axis. The solver first finds the roots of the corresponding equation ax2+bx+c=0ax^2 + bx + c = 0 using the discriminant and the quadratic formula. The roots split the number line into intervals; the solver tests one point in each interval to determine the sign of the quadratic there, then selects the intervals matching the inequality direction. The parabola's opening direction (up if a>0a > 0, down if a<0a < 0) is taken into account. Edge cases — repeated roots, no real roots — are handled by inspecting whether the parabola ever crosses zero.
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Rational
Rational Inequality Solver
P(x)Q(x)  <  >  ≤  ≥  0\dfrac{P(x)}{Q(x)} \;{<}\;{>}\;{\leq}\;{\geq}\; 0
A rational inequality is one comparing a fraction P(x)Q(x)\dfrac{P(x)}{Q(x)} to zero (or to another expression, which can be moved to one side). The standard mistake is multiplying both sides by the denominator — which is invalid here, because the sign of Q(x)Q(x) may flip between intervals. The solver instead uses a sign chart: it combines everything into a single fraction, finds every critical point (zeros of both numerator and denominator), tests one point in each resulting interval to record the sign, and selects the intervals matching the inequality. Numerator zeros are included for ≤\leq/≥\geq; denominator zeros are always excluded (they make the fraction undefined). The final answer is given in interval notation, with the union symbol ∪\cup for disjoint intervals.
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Radical
Radical Inequality Solver
P(x)n  <  >  ≤  ≥  …\sqrt[n]{P(x)} \;{<}\;{>}\;{\leq}\;{\geq}\; \dots
A radical inequality is one where the variable sits under a square root, cube root, or higher root. Two domain facts shape the solution. First, even-index radicals like P(x)\sqrt{P(x)} are only defined when P(x)≥0P(x) \geq 0, so the domain restriction must be solved alongside the inequality itself. Second,  \sqrt{\,} is always non-negative, so an inequality like P(x)<c\sqrt{P(x)} < c with c≤0c \leq 0 has no solution at all. The solver isolates the radical, raises both sides to the appropriate power to remove it, solves the resulting inequality, and intersects the result with the domain. Odd-index roots (cube root, etc.) skip the non-negativity restriction since they're defined for every real input.
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Exponential
Exponential Inequality Solver
a⋅bf(x)  <  >  ≤  ≥  ca \cdot b^{f(x)} \;{<}\;{>}\;{\leq}\;{\geq}\; c
An exponential inequality has the variable in an exponent. The key fact is that bxb^x is always strictly positive — so an inequality like bf(x)<0b^{f(x)} < 0 has no solution and one like bf(x)>0b^{f(x)} > 0 holds for every real xx. After isolating the exponential term, the solver applies a logarithm to both sides. The base controls the direction: when b>1b > 1 the function is increasing and the inequality direction is preserved; when 0<b<10 < b < 1 the function is decreasing and the direction must flip. The solver tracks this and reports the final inequality in interval notation.
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Absolute Value
Absolute Value Inequality Solver
∣ax+b∣  <  >  ≤  ≥  c|ax + b| \;{<}\;{>}\;{\leq}\;{\geq}\; c
An absolute value inequality splits into one of two structurally different forms after isolating the absolute value term. Less-than form ∣A∣<c|A| < c becomes a compound inequality −c<A<c-c < A < c (the inside is sandwiched between −c-c and cc) — a single connected interval. Greater-than form ∣A∣>c|A| > c becomes a union A<−cA < -c or A>cA > c — two separate intervals stretching to ±∞\pm\infty. The solver also handles the trivial cases up front: ∣A∣<0|A| < 0 has no solution, ∣A∣>0|A| > 0 holds everywhere except where A=0A = 0, ∣A∣≤0|A| \leq 0 has the single solution where A=0A = 0, and ∣A∣≥0|A| \geq 0 holds for all real xx. Final answers always come back in interval notation.
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Logarithmic
Logarithmic Inequality Solver
log⁡b(P(x))  <  >  ≤  ≥  c\log_b(P(x)) \;{<}\;{>}\;{\leq}\;{\geq}\; c
A logarithmic inequality has the variable inside a logarithm. Two facts shape the solution. First, log⁡b(x)\log_b(x) is only defined for x>0x > 0, so the domain of every log argument must be solved alongside the inequality and intersected with the result. Second, the base controls the direction: when b>1b > 1 the log is increasing and the inequality direction is preserved when removing the log; when 0<b<10 < b < 1 the log is decreasing and the direction must flip. The solver isolates the log, converts to exponential form using log⁡b(A)≶c  ⟺  A≶bc\log_b(A) \lessgtr c \iff A \lessgtr b^c (with the direction adjusted for the base), solves the resulting algebraic inequality, and intersects with the domain restrictions. The equal-logs case log⁡b(A)≶log⁡b(B)\log_b(A) \lessgtr \log_b(B) reduces directly to A≶BA \lessgtr B (direction-adjusted) plus domain checks.
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