Before you start, make sure you know: Inequalities (Algebra 1)
A sign at a roller coaster says riders must be at least 48 inches tall and no taller than 78 inches. Chloe is 52 in. tall and Jayden is 47 in. Who gets on? There isn't just one "right" height: a whole range of heights works. An inequality describes a range like that, and in this section you'll learn to solve every type you'll meet in Algebra 2: linear, compound, quadratic, polynomial, rational, absolute value, and radical.
Inequalities and Interval Notation
An equation usually has a few solutions. An inequality usually has infinitely many: a whole stretch of the number line. We write that stretch as an interval.
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Solve the following linear inequality, expressing the solution in interval notation: $\frac{3x - 1}{2} - \frac{x + 4}{3} \leq \frac{5x}{6} + 1$
Solve the following linear inequality, expressing the solution in interval notation:
Often two conditions must hold at the same time, like the roller coaster's height limits. That is a system (or compound inequality joined by "and"). Solve each inequality separately, then keep only the values that satisfy both: the intersection of the solution sets.
If the word is "or", you keep values that satisfy at least one: the union, written with $\cup$.
A double inequality like $-3 < 2x + 1 \leq 7$ is an "and" in disguise. Do the same operation to all three parts: subtract $1$ to get $-4 < 2x \leq 6$, then divide by $2$ to get $-2 < x \leq 3$.
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Solve the following system of linear inequalities with one unknown: $2x - 3 < 5$ $4 - x \leq 2$
Solve the following system of linear inequalities with one unknown:
A vehicle rental company offers two plans. Plan A charges a flat fee of $50 plus $0.20 per mile driven. Plan B charges a flat fee of $30 plus $0.35 per mile driven. Set…
A vehicle rental company offers two plans. Plan A charges a flat fee of $50 plus $0.20 per mile driven. Plan B charges a flat fee of $30 plus $0.35 per mile driven. Set up an inequality and determine for how many miles Plan A becomes strictly cheaper.
For a certain chemical compound to remain stable, its temperature $T$ (in degrees Fahrenheit) must simultaneously satisfy two conditions: double the temperature…
For a certain chemical compound to remain stable, its temperature $T$ (in degrees Fahrenheit) must simultaneously satisfy two conditions: double the temperature decreased by 5 degrees must not exceed 35 degrees, and a third of the temperature increased by 10 degrees must be strictly greater than 15 degrees. Set up a system of inequalities and find the temperature range in which the compound remains stable.
You can't solve $x^2 - x - 6 \leq 0$ by "isolating $x$". Instead, use the shape of the graph: the inequality asks where the parabola is below or on the $x$-axis.
When the quadratic doesn't factor, find the zeros with the quadratic formula and use the same chart. If it has no real zeros, the sign never changes: for instance $x^2 + 1 > 0$ is true for all real numbers, and $x^2 + 1 < 0$ has no solution.
Business problems use the same idea: profit is revenue minus cost, $P(x) = R(x) - C(x)$, and "profit exceeds a goal" becomes a quadratic inequality $P(x) > \text{goal}$.
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Find the solution set of the following quadratic inequality: $-2x^2 + 8x - 6 > 0$
Find the solution set of the following quadratic inequality:
The height $h$ (in feet) of a water rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine the time interval during…
The height $h$ (in feet) of a water rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine the time interval during which the rocket is at a height strictly greater than 17 feet.
A technology company manufactures solar panels. The daily production cost (in dollars) when manufacturing $x$ panels is $C(x) = 200 + 10x$. The revenue from the sale of…
A technology company manufactures solar panels. The daily production cost (in dollars) when manufacturing $x$ panels is $C(x) = 200 + 10x$. The revenue from the sale of $x$ panels is given by the function $I(x) = 50x - 0,5x^2$. Set up an inequality to determine how many panels must be manufactured and sold daily for the profits to exceed $\$400.$
The sign chart works for any degree. If the polynomial is factored, each factor gives one critical value, and you multiply the signs.
When the polynomial isn't factored, find a zero by testing small numbers like $\pm 1, \pm 2, \pm 3$ (divisors of the constant term). Once $x = r$ is a zero, use synthetic division by $x - r$ to lower the degree.
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Solve the following higher degree polynomial inequality, which is already factored: $(x - 1)(x + 2)(x - 4) \leq 0$
Solve the following higher degree polynomial inequality, which is already factored:
Find the solution set of the following fourth degree polynomial inequality. (Hint: use synthetic division to factor the polynomial). $x^4 - x^3 - 7x^2 + x + 6 < 0$
Find the solution set of the following fourth degree polynomial inequality. (Hint: use synthetic division to factor the polynomial).
Sometimes the inequality has a letter besides $x$, and the question is: for which values of that parameter is the inequality true for all real $x$? The graph answers this. For $ax^2 + bx + c > 0$ to hold everywhere, the parabola must open upward ($a > 0$) and never touch the $x$-axis, so the discriminant $D = b^2 - 4ac$ must be negative.
When the leading coefficient is also the parameter, you must split into cases, because the equation might not be quadratic at all.
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Determine the value or values of the parameter $k$ so that the quadratic inequality $x^2 - 6x + k > 0$ is satisfied for any real number $x$.
Determine the value or values of the parameter $k$ so that the quadratic inequality $x^2 - 6x + k > 0$ is satisfied for any real number $x$.
Solve the following rational inequality. Keep in mind that the polynomials in the numerator and denominator may have common roots that simplify the expression, but you…
Solve the following rational inequality. Keep in mind that the polynomials in the numerator and denominator may have common roots that simplify the expression, but you must respect the original domain:
Rates give rational inequalities. Time $=$ distance $\div$ rate, so a time condition leads to a fraction in the speed $v$. Since a speed must be positive, $v > 0$ is part of the problem.
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A cyclist plans a training route of 60 miles. They realize that if they increase their usual average speed by 5 mph, they would take at least 1 hour less to complete the…
A cyclist plans a training route of 60 miles. They realize that if they increase their usual average speed by 5 mph, they would take at least 1 hour less to complete the route. If their usual average speed is $v$ (in mph), set up a rational inequality and solve it to determine the cyclist's possible usual average speeds.
Prove theoretically that for any real number $x > 0$, the inequality $x + \frac{1}{x} \geq 2$ is satisfied. From this result, determine the solution set of the…
Prove theoretically that for any real number $x > 0$, the inequality $x + \frac{1}{x} \geq 2$ is satisfied. From this result, determine the solution set of the inequality $\frac{x^2 + 9}{x} \geq 6$.
In a precision parts factory, the diameter $d$ of a metal cylinder must measure 45 inches, with a maximum error tolerance of 0.02 inches. Express this condition using an…
In a precision parts factory, the diameter $d$ of a metal cylinder must measure 45 inches, with a maximum error tolerance of 0.02 inches. Express this condition using an inequality with an absolute value and determine the interval of acceptable measurements for the diameter.
Absolute Value with Quadratic and Rational Expressions
The same rules work when the expression inside the bars is more complicated. Remove the bars using the rules above, solve each resulting inequality with a sign chart, and then combine: intersection for "less than", union for "greater than".
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Find the solution set of the following inequality involving an absolute value over a quadratic expression: $|x^2 - 5x + 4| > 2$
Find the solution set of the following inequality involving an absolute value over a quadratic expression:
Square roots add two new checks. First, the radicand must be $\geq 0$ (the domain). Second, the square root itself is never negative, and squaring only preserves an inequality when both sides are nonnegative: $2 < 3 \Rightarrow 4 < 9$, but $-5 < 2$ while $25 > 4$.
When two radicals appear, isolate one radical so that each side is nonnegative, square, isolate the remaining radical, and square again, checking the domain at each step.
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Determine the interval of values of $x$ that satisfy the following radical inequality, taking into account the domain of the radical: $\sqrt{2x - 4} < 4$
Determine the interval of values of $x$ that satisfy the following radical inequality, taking into account the domain of the radical:
Solve the following radical inequality. Pay special attention to the domain of definition of the square root and the conditions generated when squaring:…
Solve the following radical inequality. Pay special attention to the domain of definition of the square root and the conditions generated when squaring:
Back to the start: the roller coaster's height limits are the system $48 \leq h \leq 78$, so the allowed heights are $[48, 78]$. Chloe (52 in.) rides, Jayden (47 in.) doesn't, and you can now do the same for far more complicated conditions.
Type
Strategy
Linear
Isolate $x$; flip the sign when multiplying or dividing by a negative
Compound / system
Solve each part; "and" is the intersection, "or" is the union
Quadratic and polynomial
Zero on one side, factor, sign chart (synthetic division helps find factors)
Rational
One fraction compared with $0$; denominator zeros always excluded
Absolute value
$|u| < k$ gives an "and"; $|u| > k$ gives an "or"
Radical
Domain first, then check the sign of the right side, then square
Let's practice! 8 medium exercises in this section