Inequalities in One Variable
Valeria kicks a ball and its height (in meters) after $t$ seconds is $h(t)=-5t^2+10t$. For how long was the ball more than $3$ m above the ground? Trying random numbers won't work: you need to know for which values of $t$ an expression is positive and for which it is negative. Every inequality on this page is solved with that same idea: move everything to one side, factor or reduce to a single quotient, and study the sign interval by interval.
Review: linear inequalities
An inequality is solved like an equation, with just one new rule: if you multiply or divide by a negative number, the direction of the inequality flips.
Why? Because $2<5$, but multiplying by $-1$ gives $-2$ and $-5$, and now $-2>-5$: the order is reversed.
With denominators, multiply everything by the LCM (which is positive, so the direction doesn't change). If the problem asks for natural numbers, from all the solutions you keep only the natural numbers.
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 inequality with denominators: $\frac{x}{2} - \frac{x-1}{3} > 1$.
Solve the inequality with denominators: $\frac{x}{2} - \frac{x-1}{3} > 1$.
Solve the following inequality, paying attention to the negative signs in front of the fractions: $\frac{2x - 1}{4} - \frac{3x + 2}{6} \geq \frac{x - 5}{3} - 1$.
Solve the following inequality, paying attention to the negative signs in front of the fractions: $\frac{2x - 1}{4} - \frac{3x + 2}{6} \geq \frac{x - 5}{3} - 1$.
Quadratic inequalities
A quadratic inequality is written as $ax^2+bx+c\lessgtr0$. The graph of $y=ax^2+bx+c$ is a parabola, and the inequality asks where that parabola lies above or below the $x$-axis.
Cases worth recognizing:
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Incomplete $x^2-16<0$: you solve $x^2<16$ and the roots are $\pm4$, so the solution is $(-4,4)$.
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A single root (double root): $(x-2)^2\leq0$ is only true at $x=2$, because a square is never negative.
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No real roots: $x^2+1>0$ is true for every real number; $x^2+1<0$ is never true.
Find the values of $x$ that satisfy the quadratic inequality, which is already given in factorised form: $(x - 4)(x + 2) < 0$.
Find the values of $x$ that satisfy the quadratic inequality, which is already given in factorised form: $(x - 4)(x + 2) < 0$.
Find the solution interval of the quadratic inequality: $x^2 - 5x + 6 \leq 0$.
Find the solution interval of the quadratic inequality: $x^2 - 5x + 6 \leq 0$.
Solve the following incomplete quadratic inequality and represent its solution on the number line: $2x^2 - 18 \geq 0$.
Solve the following incomplete quadratic inequality and represent its solution on the number line: $2x^2 - 18 \geq 0$.
Systems of inequalities
Geometric conditions are also systems. For a triangle with sides $x$, $x$, and $8$, the sum of any two sides must be greater than the third: $x+x>8$. If in addition the perimeter cannot exceed $30$ cm: $2x+8\leq30$. Solving, $x>4$ and $x\leq11$, so $x\in(4,11]$. (The condition $x+8>x$ always holds.)
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:
$2x - 3 < 5$
$3x + 1 \geq -2$
Solve the following system of inequalities and explain the special feature of its solution set: $x^2 - 4x + 4 \leq 0$ $2x - 1 > 0$
Solve the following system of inequalities and explain the special feature of its solution set:
$x^2 - 4x + 4 \leq 0$
$2x - 1 > 0$
Study and solve the following system consisting of a linear inequality and a quadratic inequality: $3x - 5 \leq x + 1$ $x^2 - 7x + 10 > 0$
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: factor 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: $x^3 - 4x^2 - 5x \leq 0$.
The monthly profit of an electric scooter factory, expressed in thousands of pesos, is given by the function $B(x) = -x^3 + 10x^2 - 16x$, where $x$ represents the number…
The monthly profit of an electric scooter factory, expressed in thousands of pesos, is given by the function $B(x) = -x^3 + 10x^2 - 16x$, where $x$ represents the number of hundreds of scooters manufactured. How many scooters must be manufactured so that the company does not make a loss (that is, so that $B(x) \geq 0$)?
Rational: sign of the numerator and the denominator
When there are fractions on both sides, you have to combine them.
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: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.
Solve the following rational inequality, simplifying the algebraic additions required to express it as a single quotient before studying the signs:…
Solve the following rational inequality, simplifying the algebraic additions required to express it as a single quotient before studying the signs: $\frac{1}{x} + \frac{1}{x+2} \geq \frac{3}{4}$.
Absolute value
$|x-a|$ is the distance from $x$ to $a$ on the number line. Therefore:
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$|x-a|
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$|x-a|>k$ means "more than $k$ away from $a$": two rays, $x-a>k$ or $x-a<-k$.
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: $\left| \frac{3x - 1}{2} \right| < 4$.
Word problems with linear inequalities
To set up the problem, define the variable, translate the words ("at least" is $\geq$, "less than" is $<$, "must not exceed" is $\leq$), and at the end check that the solution makes sense (for example, that it is a whole number).
The sum of double a natural number and $5$ is strictly less than $17$. Set up the corresponding inequality and find all the natural numbers that satisfy it.
The sum of double a natural number and $5$ is strictly less than $17$. Set up the corresponding inequality and find all the natural numbers that satisfy it.
A small craft business has monthly fixed costs of $6\,000$ pesos and producing each leather bracelet costs $40$ pesos. If it sells each bracelet for $100$ pesos, how…
A small craft business has monthly fixed costs of $6\,000$ pesos and producing each leather bracelet costs $40$ pesos. If it sells each bracelet for $100$ pesos, how many bracelets must it sell per month as a minimum to start making a profit? Set up and solve the inequality.
A delivery van consumes $8$ liters of fuel every $100 \text{ km}$ on the highway and $12$ liters every $100 \text{ km}$ in the city. If on a trip of $500 \text{ km}$ in…
A delivery van consumes $8$ liters of fuel every $100 \text{ km}$ on the highway and $12$ liters every $100 \text{ km}$ in the city. If on a trip of $500 \text{ km}$ in total it has consumed less than $45$ liters, what is the maximum number of kilometers it could have driven in the city?
In an isosceles triangle, the equal sides measure $x \text{ cm}$ and the third side measures $10 \text{ cm}$. For the triangle to be constructed, the sum of the lengths…
In an isosceles triangle, the equal sides measure $x \text{ cm}$ and the third side measures $10 \text{ cm}$. For the triangle to be constructed, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Furthermore, its perimetre must not exceed $36 \text{ cm}$. Set up a system of inequalities and find the possible values of $x$.
Quadratic word problems and parameters
When motion gives the inequality. For Valeria's ball we want $-5t^2+10t>3$. We move everything to one side: $-5t^2+10t-3>0$. We multiply by $-1$ (and the direction flips): $5t^2-10t+3<0$. With the quadratic formula the roots are $t=1\pm\dfrac{\sqrt{10}}{5}\approx0.37$ and $1.63$. The parabola is negative between them, so the ball is more than $3$ m high from $t\approx0.37$ s to $t\approx1.63$ s: about $1.26$ s. That answers the question at the start.
Parameters. If the problem asks "for which values of $k$…?", you almost always have to use the discriminant $\Delta=b^2-4ac$ of a quadratic equation:
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$\Delta<0$: no real solutions.
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$\Delta=0$: one double solution.
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$\Delta>0$: two distinct real solutions.
That condition is an inequality in $k$.
A rectangular garden is to be built such that its length is $4$ meters greater than its width $x$. If there is enough fertilizer to cover a maximum of $45 \text{ m}^2$,…
A rectangular garden is to be built such that its length is $4$ meters greater than its width $x$. If there is enough fertilizer to cover a maximum of $45 \text{ m}^2$, set up an inequality and determine the possible values that the width $x$ can take.
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.
A water rocket is launched vertically upward in a school experiment. Its height $h$ in meters as a function of time $t$ in seconds is given by the expression…
A water rocket is launched vertically upward in a school experiment. Its height $h$ in meters as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine algebraically during which time interval the rocket will be above a height of $17 \text{ m}$.