Inequalities in one unknown
Beyond first and second degree there are inequalities with $x^3$, with $x$ in the denominator or hidden inside an absolute value. They are all solved with the same idea: move everything to one side, factorise or simplify to a single quotient, and study the sign interval by interval.
Mixed systems: first and second degree together
Solve the following linear inequality: $3(x - 2) + 4 \leq 5x - 8$. Express the solution in interval notation.
Solve the following linear inequality: $3(x - 2) + 4 \leq 5x - 8$. Express the solution in interval notation.
Solve the following system of inequalities with one unknown and express the solution as a single interval: $2x - 3 < 5$ $3x + 1 \geq -2$
Solve the following system of inequalities with one unknown and express the solution as a single interval:
$2x - 3 < 5$
$3x + 1 \geq -2$
Study and solve the following system consisting of a linear inequality and a quadratic inequality: $3x - 5 \leq x + 1$ $x^2 - 7x + 10 > 0$
Study and solve the following system consisting of a linear inequality and a quadratic inequality:
$3x - 5 \leq x + 1$
$x^2 - 7x + 10 > 0$
Higher degree: factorising before studying the sign
Solve the higher-degree polynomial inequality by first taking out a common factor: $x^3 - 4x^2 - 5x \leq 0$.
Solve the higher-degree polynomial inequality by first taking out a common factor: $x^3 - 4x^2 - 5x \leq 0$.
Rational inequalities: studying the sign of numerator and denominator
Solve the simple rational inequality by studying the signs of the numerator and denominator: $\frac{x-3}{x+1} \leq 0$.
Solve the simple rational inequality by studying the signs of the numerator and denominator: $\frac{x-3}{x+1} \leq 0$.
Solve the rational inequality: $\frac{2x + 1}{x - 2} > 3$. Remember to move all terms to one side and simplify to obtain a single fraction before studying the signs.
Solve the rational inequality: $\frac{2x + 1}{x - 2} > 3$. Remember to move all terms to one side and simplify to obtain a single fraction before studying the signs.
Solve the rational inequality where the numerator and the denominator are quadratic polynomials: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.
Solve the rational inequality where the numerator and the denominator are quadratic polynomials: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.
Inequalities with absolute value
Solve the inequality with modulus and express its solution as a union of intervals: $|2x - 5| \geq 7$.
Solve the inequality with modulus and express its solution as a union of intervals: $|2x - 5| \geq 7$.
Solve the modulus inequality that includes fractional expressions: $\left| \frac{3x - 1}{2} \right| < 4$.
Solve the modulus inequality that includes fractional expressions: $\left| \frac{3x - 1}{2} \right| < 4$.
Problems and parameters
When the problem asks "for which values of $k$...?", the condition usually depends on the discriminant $\Delta=b^2-4ac$ of a quadratic equation: $\Delta<0$ (no real solutions), $\Delta=0$ (one repeated solution) or $\Delta>0$ (two solutions), and that in turn becomes an inequality in $k$.
A rectangular garden is to be built such that its length is $4$ metres greater than its width $x$. If there is enough fertiliser to cover a maximum of $45 \text{ m}^2$,…
A rectangular garden is to be built such that its length is $4$ metres greater than its width $x$. If there is enough fertiliser to cover a maximum of $45 \text{ m}^2$, set up an inequality and determine the possible values that the width $x$ can take.
A small craft business has monthly fixed costs of $300$ euros and producing each leather bracelet costs $2$ euros. If it sells each bracelet for $5$ euros, how many…
A small craft business has monthly fixed costs of $300$ euros and producing each leather bracelet costs $2$ euros. If it sells each bracelet for $5$ euros, how many bracelets must it sell per month as a minimum to start making a profit? Set up and solve the inequality.
Determine for which values of the parameter $k$ the quadratic equation $x^2 - kx + 9 = 0$ has no real solutions. Set up the corresponding inequality using the…
Determine for which values of the parameter $k$ the quadratic equation $x^2 - kx + 9 = 0$ has no real solutions. Set up the corresponding inequality using the discriminant and solve it.