Linear Inequalities in One Variable
Emma wants a pair of sneakers that cost $\$120$. She has $\$45$ and saves $\$15$ every week. After how many weeks can she buy them? Real life rarely asks for "exactly $\$120$". It asks for at least, at most, cheaper than. That is the job of an inequality. You solve it almost like an equation, with one new rule that changes everything: multiplying or dividing by a negative number flips the sign.
Solutions and graphs
An equation like $x + 1 = 4$ has one solution. An inequality like $x + 1 < 4$ has infinitely many: $2$, $0$, $-7$, $2.9$, and so on. So we describe the solution set with a graph or with interval notation.
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Open circle and parenthesis for $<$ or $>$: the endpoint is not included.
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Closed circle and bracket for $\le$ or $\ge$: the endpoint is included.
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$\infty$ and $-\infty$ always get a parenthesis, because they are not numbers you can reach.
Given the inequality $4x - 7 \ge 2x + 1$, check which of the following values belong to its solution set: $x = -2$, $x = 0$, $x = 4$, $x = 5$.
Given the inequality $4x - 7 \ge 2x + 1$, check which of the following values belong to its solution set: $x = -2$, $x = 0$, $x = 4$, $x = 5$.
Solving: inverse operations and the flip rule
You isolate $x$ with inverse operations, doing the same thing to both sides. Adding, subtracting, and multiplying or dividing by a positive number never change the direction. Multiplying or dividing by a negative number does.
Solve the following linear inequality: $3x - 5 < 2x + 4$
Solve the following linear inequality:
$3x - 5 < 2x + 4$
Find the values of $x$ that satisfy the inequality: $4 - 3x > 10$
Find the values of $x$ that satisfy the inequality:
$4 - 3x > 10$
Simplify and solve the linear inequality: $x + 2x + 3x < 12 + 2x$
Simplify and solve the linear inequality:
$x + 2x + 3x < 12 + 2x$
Parentheses and fractions
Clean the inequality up first:
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Distribute to remove parentheses.
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Clear fractions by multiplying every term by the least common denominator (LCD). The LCD is positive, so the sign does not flip.
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Combine like terms, then isolate $x$ as before.
Find the solution set of the inequality by first removing parentheses: $2(x - 3) + 4 \ge 5x - 1$
Find the solution set of the inequality by first removing parentheses:
$2(x - 3) + 4 \ge 5x - 1$
Solve the following inequality with denominators: $\frac{x}{2} - 1 \le \frac{3x}{4} + 2$
Solve the following inequality with denominators:
$\frac{x}{2} - 1 \le \frac{3x}{4} + 2$
Solve the following inequality combining the use of parentheses and denominators: $\frac{2(x-1)}{3} - \frac{3x+1}{4} \le \frac{5-x}{6}$
Solve the following inequality combining the use of parentheses and denominators:
$\frac{2(x-1)}{3} - \frac{3x+1}{4} \le \frac{5-x}{6}$
Decimals and whole-number answers
Decimals are fractions in disguise. Multiply every term by $10$ or $100$ (positive!) to turn them into whole numbers.
Sometimes a problem asks only for whole numbers, such as "the smallest integer $x$ such that…". Solve first, then pick the right integer from the solution set. Never round.
Solve the following inequality by first working with decimal numbers: $0.5(x - 2) + 0.25x \ge 1.5x - 0.75$
Solve the following inequality by first working with decimal numbers:
$0.5(x - 2) + 0.25x \ge 1.5x - 0.75$
Determine the smallest integer $x$ that satisfies the following inequality: $5x + 7 > 22$
Determine the smallest integer $x$ that satisfies the following inequality:
$5x + 7 > 22$
Translating words into inequalities
Solving is half the job. The other half is reading the problem and writing the inequality. Use the same four steps every time.
| Words | Symbol |
|---|---|
| less than, fewer than, below | $<$ |
| at most, no more than, cannot exceed, maximum | $\le$ |
| greater than, more than, exceeds, above | $>$ |
| at least, no less than, minimum | $\ge$ |
| between 2 and 9 | $2 < x < 9$ |
For example, "three times a number plus $4$ is at most $25$" is $3n + 4 \le 25$.
Translate into algebraic language and solve: "Twice a number minus three is strictly less than the number itself plus five."
Translate into algebraic language and solve: "Twice a number minus three is strictly less than the number itself plus five."
In a rectangle, it is known that the base measures $5$ in. more than the height. If the perimeter of the rectangle must be strictly less than $50$ in., what are the…
In a rectangle, it is known that the base measures $5$ in. more than the height. If the perimeter of the rectangle must be strictly less than $50$ in., what are the possible values that the height measurement can take?
A freight elevator has a maximum capacity of $800$ pounds. A worker weighing $75$ pounds must take up boxes of building materials weighing $22$ pounds each. If the…
A freight elevator has a maximum capacity of $800$ pounds. A worker weighing $75$ pounds must take up boxes of building materials weighing $22$ pounds each. If the worker must always accompany the load in the elevator, set up and solve an inequality to determine the maximum number of whole boxes they can transport in a single trip.
Comparing plans: which is cheaper?
To find when one option beats another, write both costs as expressions in the same variable and set up an inequality between them.
On the graph, each plan is a line, and the lower line is the cheaper plan. The lines cross at the break-even point, where the costs are equal. The inequality's solution is the set of inputs where Gym A's line is below Gym B's line, to the right of the crossing.
A car rental company (Co. A) charges a flat fee of $\$30$ plus $\$0.15$ per mile traveled. Another competing company (Co. B) charges a flat $\$15$ plus $\$0.20$ per…
A car rental company (Co. A) charges a flat fee of $\$30$ plus $\$0.15$ per mile traveled. Another competing company (Co. B) charges a flat $\$15$ plus $\$0.20$ per mile. Set up an inequality to find out at how many miles it becomes cheaper to rent the car from Co. A and interpret the result using the following graph.
A telecommunications company offers two data plans. The Standard Plan costs $\$10$ per month and includes $2$ GB of data, charging $\$2$ for each additional GB. The Plus…
A telecommunications company offers two data plans. The Standard Plan costs $\$10$ per month and includes $2$ GB of data, charging $\$2$ for each additional GB. The Plus Plan costs $\$20$ per month and includes $5$ GB of data, charging $\$1.5$ for each additional GB. If a user knows they always consume more than $5$ GB per month, what is the minimum number of gigabytes they must consume for the Plus Plan to be strictly cheaper than the Standard Plan?
Profit, averages and formulas
Three setups appear again and again.
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Profit is revenue minus cost: $P = R - C$.
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A mean of $n$ scores is (sum of scores) $\div\, n$. To reach a target mean, put the unknown score into the sum.
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A formula such as $F = \frac95 C + 32$ or $d = 55t$ turns into an inequality when you substitute the limit for one variable, such as "$d \ge 220$", and solve for the other.
The sloped line is the profit and the flat line is the $\$300$ goal. The solution is where profit is on or above the goal.
A small company manufactures cell phone cases. The fixed monthly costs are $\$500$ and producing each case costs $\$2$. If each case is sold for $\$5…
A small company manufactures cell phone cases. The fixed monthly costs are $\$500$ and producing each case costs $\$2$. If each case is sold for $\$5$, set up an inequality and determine the minimum number of cases that must be sold per month to obtain a profit of at least $\$1000$.
A high school student has obtained the following grades on their first three math exams: $6.5$, $7.0$, and $8.2$. What minimum grade must they get on the fourth exam for…
A high school student has obtained the following grades on their first three math exams: $6.5$, $7.0$, and $8.2$. What minimum grade must they get on the fourth exam for their average grade to be greater than or equal to $7.5$?
The relationship between the temperature in degrees Fahrenheit ($F$) and degrees Celsius ($C$) is given by the formula $F = \frac{9}{5}C + 32$. Set up an inequality and…
The relationship between the temperature in degrees Fahrenheit ($F$) and degrees Celsius ($C$) is given by the formula $F = \frac{9}{5}C + 32$. Set up an inequality and solve it to determine from what temperature in degrees Celsius $86$ degrees Fahrenheit is exceeded.
Mixtures and investments
In these problems, track the amount of the important stuff (acid, antifreeze, interest) and the total. When you multiply by a total such as $8 + x$, it is positive, so the sign stays.
A laboratory has $10$ liters of an acid solution at $20\%$. Set up an inequality to calculate how many liters of pure acid (at $100\%$) must be added so that the…
A laboratory has $10$ liters of an acid solution at $20\%$. Set up an inequality to calculate how many liters of pure acid (at $100\%$) must be added so that the concentration of the new mixture is at least $50\%$.
An investor has $\$20000$ in capital. They decide to invest part in a low-risk fund that offers a $4\%…
An investor has $\$20000$ in capital. They decide to invest part in a low-risk fund that offers a $4\%$ annual return, and the rest in a technology fund that offers a $6\%$ annual return. What is the minimum amount they must invest in the technology fund to ensure that the total interest generated after one year is at least $\$1000$?
Inequalities with a parameter
When an inequality contains another letter besides $x$ (a parameter), you cannot always divide by its coefficient. The sign of the coefficient decides the direction, and the coefficient might be $0$.
Parameters also appear in reverse: "for what value of the parameter is the solution set exactly ...?" Solve in terms of the parameter, then match the boundary.
Consider the inequality $3x - a > 2x + 5$, where $a$ is a real parameter. Calculate the value of $a$ knowing that the solution set of the inequality is the interval…
Consider the inequality $3x - a > 2x + 5$, where $a$ is a real parameter. Calculate the value of $a$ knowing that the solution set of the inequality is the interval $(8, \infty)$.
Determine the value of the parameter $m$ so that the solution set of the inequality $\frac{mx - 1}{2} - \frac{x + m}{3} > 1$ is exactly the interval $(4, \infty)$.
Determine the value of the parameter $m$ so that the solution set of the inequality
$\frac{mx - 1}{2} - \frac{x + m}{3} > 1$
is exactly the interval $(4, \infty)$.
Analyze and solve the inequality $kx - 3 \le 2x + k$ in terms of the possible values of the real parameter $k$.
Analyze and solve the inequality $kx - 3 \le 2x + k$ in terms of the possible values of the real parameter $k$.
Literal inequalities and justifying your answer
In a literal inequality the coefficients are letters, but the problem gives you their signs (such as $a > b > 0$). Use those facts to decide whether to flip the sign.
To justify a result analytically, do three things:
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Write each step and the property you used (for example, "multiply by $6$, which is positive, so the sign stays").
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State the solution set clearly.
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Check a value at the boundary, a value inside the set, and a value outside the set.
Solve the following literal inequality for the unknown $x$, assuming that the parameters satisfy the condition $a > b > 0$:…
Solve the following literal inequality for the unknown $x$, assuming that the parameters satisfy the condition $a > b > 0$:
$\frac{x}{a} - \frac{x}{b} \ge \frac{a^2 - b^2}{ab}$
Solve the following inequality and justify the result obtained analytically: $\frac{3(x+2)}{4} - \frac{x-1}{2} \ge \frac{x+5}{4} + 1$
Solve the following inequality and justify the result obtained analytically:
$\frac{3(x+2)}{4} - \frac{x-1}{2} \ge \frac{x+5}{4} + 1$
Back to the start: Emma needs $45 + 15w \ge 120$. Subtract $45$ to get $15w \ge 75$, and divide by $15$ to get $w \ge 5$. She can buy the sneakers after $5$ weeks, with exactly $\$120$ saved.