Equations and Systems · Equations, Inequalities and Systems
Solving Inequalities in One Variable
Aaliyah throws a ball straight up. Its height in feet after $t$ seconds is $h = -16t^2 + 40t + 4$. For how long is the ball higher than 28 ft, where a drone camera is filming? That is an inequality, not an equation, and it has a whole set of answers. On this page you will build one toolkit for every kind of one-variable inequality: linear, quadratic, higher degree, rational and absolute value.
Linear inequalities
A linear inequality is solved like a linear equation, with one new rule. The answer is usually a whole interval of numbers, so we graph it on a number line and write it in interval notation.
An open circle (or parenthesis) means the endpoint is not included, so it goes with $<$ and $>$. A closed circle (or bracket) means it is included, so it goes with $\le$ and $\ge$. Infinity always gets a parenthesis.
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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 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: $\frac{2x - 1}{4} - \frac{3x + 2}{6} \geq \frac{x - 5}{3} - 1$.
For $x^2 - x - 6 > 0$ the left side is a parabola. We want the $x$-values where the parabola is above the $x$-axis. A smooth graph can only change from positive to negative by crossing the axis, so the zeros split the number line into parts where the sign never changes.
A quadratic with no $x$-term, like $x^2 \ge 9$, is solved without factoring by isolating $x^2$. Some problems call it an incomplete quadratic. The zeros are $x = \pm 3$.
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Find the values of $x$ that satisfy the quadratic inequality, which is already factored: $(x - 4)(x + 2) < 0$.
Find the values of $x$ that satisfy the quadratic inequality, which is already factored: $(x - 4)(x + 2) < 0$.
The monthly profits of an electric scooter factory, expressed in thousands of dollars, are given by the function $B(x) = -x^3 + 10x^2 - 16x$, where $x$ represents the…
The monthly profits of an electric scooter factory, expressed in thousands of dollars, are 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 $B(x) \geq 0$)?
When $x$ is in a denominator, the sign of the fraction depends on the sign of the top and the bottom. Signs of quotients follow the same rules as signs of products.
Look at the graph of $y = \dfrac{x-1}{x+3}$. It touches the $x$-axis at $x = 1$ and has a vertical break at $x = -3$. It is above the axis on both outer parts and below it in between.
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Solve the simple rational inequality by studying the sign of the numerator and the denominator: $\frac{x-3}{x+1} \leq 0$.
Solve the simple rational inequality by studying the sign of the numerator and the denominator: $\frac{x-3}{x+1} \leq 0$.
Solve the rational inequality: $\frac{2x + 1}{x - 2} > 3$. Remember to move all the terms to one side and operate to obtain a single quotient before studying the signs.
Solve the rational inequality: $\frac{2x + 1}{x - 2} > 3$. Remember to move all the terms to one side and operate to obtain a single quotient before studying the signs.
Sometimes the numerator and denominator are quadratics, or the inequality has several fractions that you must add first. The steps are the same: one fraction on one side, $0$ on the other, factor, then make a sign chart.
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Solve the rational inequality in which the numerator and the denominator are quadratic polynomials: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.
Solve the rational inequality in which the numerator and the denominator are quadratic polynomials: $\frac{x^2 - 2x - 8}{x^2 - 9} \leq 0$.
Solve the following rational inequality, performing the necessary algebraic sums to express it as a single quotient before studying the signs:…
Solve the following rational inequality, performing the necessary algebraic sums to express it as a single quotient before studying the signs: $\frac{1}{x} + \frac{1}{x+2} \geq \frac{3}{4}$.
The absolute value $|x - 4|$ is the distance from $x$ to $4$ on the number line. So $|x - 4| < 3$ asks for the numbers within $3$ units of $4$, and $|x - 4| > 3$ asks for the numbers farther than $3$ units away.
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Solve the absolute value inequality and express its solution as a union of intervals: $|2x - 5| \geq 7$.
Solve the absolute value inequality and express its solution as a union of intervals: $|2x - 5| \geq 7$.
A system asks for all its conditions at the same time. Solve each inequality separately, then take the intersection: the numbers that belong to every solution set. A double inequality like $-2 \le 3x + 1 < 7$ is also a system, "$-2 \le 3x + 1$ and $3x + 1 < 7$". Solve it by working on all three parts: $-3 \le 3x < 6$, so $-1 \le x < 2$.
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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:
To solve a word problem, define the variable, translate the words into an inequality, solve it, and then ask whether the answer makes sense (whole numbers, positive amounts).
Words
Symbol
more than, exceeds, greater than
$>$
at least, minimum, no less than
$\ge$
less than, fewer than, under
$<$
at most, maximum, no more than, does not exceed
$\le$
If the unknown must be a natural number ($1, 2, 3, \dots$), list the values that fit your interval. For example, if $n < 4.5$, then $n = 1, 2, 3, 4$.
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The sum of twice 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 twice 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 fixed monthly costs of $\$300$ and producing each leather bracelet costs $\$2$. If it sells each bracelet for $\$5…
A small craft business has fixed monthly costs of $\$300$ and producing each leather bracelet costs $\$2$. If it sells each bracelet for $\$5$, what is the minimum number of bracelets it must sell per month to start making a profit? Set up and solve the inequality.
A delivery van consumes $8$ gallons of fuel per $100 \text{ mi}$ on the highway and $12$ gallons per $100 \text{ mi}$ in the city. If on a trip of $500 \text{ mi}$ in…
A delivery van consumes $8$ gallons of fuel per $100 \text{ mi}$ on the highway and $12$ gallons per $100 \text{ mi}$ in the city. If on a trip of $500 \text{ mi}$ in total it has consumed less than $45$ gallons, what is the maximum number of miles it could have traveled in the city?
Quadratic models describe thrown objects, areas and profits. Set up the inequality, solve it with the zeros of a parabola, and then keep only the answers that make sense in context.
Back to the start: Aaliyah's ball stays above 28 ft for only half a second, between $t = 1$ and $t = 1.5$ seconds, so the drone has to be quick.
Some problems give several conditions at once, which makes a system. For the sides of a triangle the triangle inequality says the sum of any two sides must be strictly greater than the third.
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A rectangular garden is to be built so that its length is $4$ feet greater than its width $x$. If there is enough fertilizer to cover a maximum of $45 \text{ ft}^2$, set…
A rectangular garden is to be built so that its length is $4$ feet greater than its width $x$. If there is enough fertilizer to cover a maximum of $45 \text{ ft}^2$, set up an inequality and determine the possible values that the width $x$ can take.
A water rocket is launched vertically upwards in a school experiment. Its height $h$ in feet as a function of time $t$ in seconds is given by the expression…
A water rocket is launched vertically upwards in a school experiment. Its height $h$ in feet 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 $17 \text{ ft}$ in height.
In an isosceles triangle, the equal sides measure $x \text{ cm}$ and the unequal side measures $10 \text{ cm}$. For the triangle to be constructed, the sum of the…
In an isosceles triangle, the equal sides measure $x \text{ cm}$ and the unequal 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, it is desired that its perimeter does not exceed $36 \text{ cm}$. Set up a system of inequalities and find the possible values of $x$.
Sometimes a problem asks, "For which values of $k$ does the equation have no real solutions?" The answer lies in the discriminant $\Delta = b^2 - 4ac$ of $ax^2 + bx + c = 0$.
$\Delta > 0$: two real solutions.
$\Delta = 0$: one real solution (a double root).
$\Delta < 0$: no real solutions.
Turn the condition into an inequality in $k$ and solve it with the methods above.
Drag $k$. For small $|k|$ the parabola floats above the $x$-axis. At some value it just touches the axis, and past that it crosses it twice.
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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.