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Solved like equations, with one twist that changes everything.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you solve inequalities such as $3x + 4 < 19$ with the same two moves you use for equations, and you learn the one twist: multiplying or dividing both sides by a negative number reverses the sign. That is not an arbitrary rule ($2 < 3$ but $-2 > -3$), and you will see why. The answer is a whole range of numbers rather than one value, so you graph it and say what it means in the story.
You can solve two-step equations such as $3x + 5 = 20$ by undoing the operations in reverse order, doing the same thing to both sides. You know the inequality symbols: $<$ means less than, $>$ means greater than, $\le$ means less than or equal to, and $\ge$ means greater than or equal to. You can graph a simple inequality such as $x > 2$ on a number line, and you can multiply and divide with negative numbers. In this lesson you put these together to solve inequalities that need two steps.
| Term | What it means |
|---|---|
| Inequality | A statement that one quantity is less than, greater than, or not equal to another, such as $2x + 1 < 9$. |
| Solution set | All the numbers that make an inequality true. It is usually a whole range of numbers, not one value. |
| Boundary | The number where the solution set starts or stops, found by solving as if the sign were an equals sign. |
| Open circle | A circle on a number line showing that the boundary is not in the solution set, used for $<$ and $>$. |
| Filled circle | A circle showing that the boundary is in the solution set, used for $\le$ and $\ge$. |
| At least, at most | At least 10 means $\ge 10$; at most 10 means $\le 10$. |
| Reverse the sign | Turn $<$ into $>$ (or $\le$ into $\ge$), which you must do when you multiply or divide both sides by a negative number. |
A two-step inequality such as $2x + 3 < 11$ is solved with the same moves as the equation $2x + 3 = 11$: undo the constant, then undo the coefficient, doing the same thing to both sides.
The answer is not one number but a solution set: every number less than 4 works, such as 3, 0, $-10$ and 3.99. On a number line it is an open circle at 4 with everything to its left shaded.
The twist is this: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. Adding or subtracting never changes the sign, and multiplying or dividing by a positive number never changes it, but a negative multiplier or divisor always does.
Another way: picture
Mark 2 and 3 on a number line: 2 is to the left, so $2 < 3$. Now multiply both by $-1$. The points jump to $-2$ and $-3$, mirror images across 0, and now $-2$ is to the right of $-3$. Multiplying by a negative number flips the whole line, so the order of any two numbers flips too, and the sign must flip with it.
Another way: story
Two divers are underwater: Ava at 2 meters deep and Ben at 3 meters. Ava's depth is less than Ben's, $2 < 3$. But their positions, measured from the surface, are $-2$ and $-3$, and Ava is higher: $-2 > -3$. Changing depths to positions multiplies by $-1$, and the comparison turns around.
Try it with numbers you can check. Start with the true statement $6 < 10$.
Only the negative multiplier and the negative divisor turned the comparison around. This is not a rule someone made up; it is what happens to the numbers, and you can always test it with a pair like 6 and 10 if you forget.
The solution set of a one-variable inequality is drawn on a number line with a circle at the boundary and an arrow of shading.
| Inequality | Circle at the boundary | Shading |
|---|---|---|
| $x < 4$ | open | left |
| $x \le 4$ | filled | left |
| $x > 4$ | open | right |
| $x \ge 4$ | filled | right |
Read the inequality with $x$ on the left: then the symbol points the same way as the shading, because $<$ opens toward the smaller numbers on the left. If your answer comes out as $4 > x$, rewrite it as $x < 4$ before you graph it.
In $-3x + 5 \ge 17$, first subtract 5 from both sides: $-3x \ge 12$. Now divide both sides by $-3$ and reverse the sign: $x \le -4$. The solution set is a filled circle at $-4$, shaded to the left.
Test it: $x = -5$ gives $-3(-5) + 5 = 20$, and $20 \ge 17$ is true. $x = 0$ gives $5 \ge 17$, which is false. So the numbers to the left of $-4$ work and the numbers to the right do not, exactly as the graph says. A learner who forgot to reverse the sign would write $x \ge -4$, and the test with $x = 0$ catches that at once.
Watch for the negative sign when the variable term is subtracted, as in $9 - 2x < 1$: the coefficient is $-2$, so the sign reverses when you divide.
Inequalities describe limits: a budget you cannot go over, a score you must reach, a weight a bridge can carry. The words tell you the symbol.
A family can spend at most 60 dollars at a fair with a 12-dollar entry fee and rides at 4 dollars each: $4r + 12 \le 60$. Then $4r \le 48$ and $r \le 12$, so they can go on at most 12 rides. Always answer the question in words.
In a story, the variable often has to be a whole number: people, tickets, weeks, books. Then the solution set is only the whole numbers in the range, and the direction of rounding comes from the inequality.
If the answer is $n \le 7.6$ (at most), the largest whole number that works is 7, so you round down. If the answer is $n \ge 7.6$ (at least), the smallest whole number that works is 8, so you round up, even though 7.6 is closer to 8 anyway in this case; $n \ge 7.2$ would still need 8. Ordinary rounding to the nearest whole number is the wrong tool here. Check the whole numbers on each side of the boundary to be sure.
It helps to solve an equation and an inequality next to each other. The equation $5x - 2 = 18$ has one solution, $x = 4$: that is the only number that makes the two sides equal. The inequality $5x - 2 > 18$ uses exactly the same steps and gives $x > 4$, which has infinitely many solutions: 5, 4.1, 100 and every other number above 4. The number 4 itself is the boundary, the place where the inequality changes from false to true.
So you can think of solving an inequality as two jobs. First find the boundary, by solving as if the sign were an equals sign. Then decide which side of the boundary the solutions are on, and whether the boundary belongs, by testing numbers. The sign-reversal rule is simply a shortcut for the second job.
Use these steps for every two-step inequality.
How to check. Solving with the sign replaced by an equals sign gives the boundary; put the boundary into the original inequality to see whether it is included. Then test one number from the shaded side, which must make the original inequality true, and one from the other side, which must make it false. Numbers that are easy to substitute, such as 0, are good test values when they are not the boundary. If the shaded side fails the test, the sign should have been reversed, or should not have been.
Most US airlines charge an extra fee for a checked bag that weighs more than 50 pounds. Suppose a packed suitcase weighs 38.5 pounds and a traveler wants to add jars of maple syrup that weigh 1.75 pounds each. The inequality is $1.75j + 38.5 \le 50$. Subtract 38.5: $1.75j \le 11.5$. Divide by 1.75: $j \le 6.57\ldots$ Only whole jars can be packed, so the traveler can add at most 6 jars. Check: $6 \times 1.75 + 38.5 = 49$ pounds, under the limit, while 7 jars would make 50.75 pounds and cost a fee.
Food safety experts say a home freezer should be kept at $0^\circ$F or colder. A new freezer is plugged in at $20^\circ$F and cools by about 4 degrees an hour. After $h$ hours its temperature is $20 - 4h$, and it is safe when $20 - 4h \le 0$. Subtract 20: $-4h \le -20$. Divide by $-4$ and reverse the sign: $h \ge 5$. So the food should go in after at least 5 hours. The test agrees: after 6 hours the temperature is $20 - 24 = -4^\circ$F, which is colder than 0.
The federal minimum wage in the United States is 7.25 dollars an hour, and it has been that since 2009. A teen who has 25 dollars and wants at least 200 dollars for a phone needs $7.25h + 25 \ge 200$. Subtracting 25 gives $7.25h \ge 175$, and dividing gives $h \ge 24.1\ldots$ Rounding up, that is at least 25 hours of work at the minimum wage. Many states set a higher minimum, and at a higher wage the same inequality gives fewer hours.
The most common mistake is forgetting to reverse the sign after dividing by a negative number. $-4x > 12$ gives $x < -3$, not $x > -3$. Test a value to catch it.
The second is reversing the sign when it should not change: after subtracting a negative number, or after dividing a negative number by a positive one. In $2x < -10$ you divide by positive 2, so $x < -5$ keeps its sign. What matters is the sign of the number you divide by, not the sign of the answer.
The third is mixing up open and filled circles. Only $\le$ and $\ge$ include the boundary.
The last is rounding the wrong way in a story. At most 7.6 books is 7 books; at least 7.6 weeks is 8 weeks.
Subtract 4 from both sides.
$3x + 4 - 4 < 19 - 4$
Subtracting the same number from both sides keeps the inequality true.
Simplify both sides.
$3x < 15$
Only the multiplication by 3 is left.
Divide both sides by 3.
$\dfrac{3x}{3} < \dfrac{15}{3}, \quad x < 5$
3 is positive, so the sign stays the same.
Test a value from the solution set.
$x = 0: \quad 3(0) + 4 = 4 < 19 \text{ true}$
Zero is less than 5, and it makes the original inequality true.
Graph the solution set.
$\text{open circle at } 5, \text{ shaded left}$
5 itself gives $19 < 19$, which is false, so the circle is open.
Subtract 7 from both sides.
$-2x + 7 - 7 \ge 15 - 7$
The constant is cleared first, and subtracting never changes the sign.
Simplify both sides.
$-2x \ge 8$
The coefficient of $x$ is $-2$.
Divide both sides by $-2$ and reverse the sign.
$\dfrac{-2x}{-2} \le \dfrac{8}{-2}$
Dividing by a negative number reverses the order, so $\ge$ becomes $\le$.
Simplify the division.
$x \le -4$
Positive 8 divided by negative 2 is $-4$.
Test a value on each side of the boundary.
$x = -5: 17 \ge 15 \text{ true}; \quad x = 0: 7 \ge 15 \text{ false}$
The shaded side works and the other side does not.
Graph the solution set.
$\text{filled circle at } -4, \text{ shaded left}$
The sign $\le$ includes the boundary, so the circle is filled.
Name the unknown.
$h = \text{hours worked}$
The question asks how many hours, so that is the variable.
Write the inequality: 40 dollars saved plus 7.25 dollars an hour, at least 150 dollars.
$7.25h + 40 \ge 150$
At least means the total may equal 150 or be more.
Subtract 40 from both sides.
$7.25h + 40 - 40 \ge 150 - 40$
The money already saved is taken off first.
Simplify both sides.
$7.25h \ge 110$
This is what the hours of work must earn.
Divide both sides by 7.25.
$h \ge \dfrac{110}{7.25}$
The hourly pay is positive, so the sign stays the same.
Work out the division.
$h \ge 15.17\ldots$
Fifteen hours is not quite enough.
Round up to whole hours.
$h = 16$
At least 15.17 hours means the smallest whole number that works is 16.
Check the whole numbers on each side.
$15: 148.75 < 150; \quad 16: 156 \ge 150$
Fifteen hours falls short and sixteen is enough, so 16 hours is the answer.
Add 3 to both sides.
$\dfrac{x}{4} - 3 + 3 \le 2 + 3, \quad \dfrac{x}{4} \le 5$
Adding undoes the subtraction and never changes the sign.
Multiply both sides by 4.
Graph and test.
Which graph shows the solution set of $x \ge -5$?
Complete the worked solution of $26 - 7x < -30$.
Subtract $26$ from both sides.
$-7x <$ p
The constant term is cleared first, just as in an equation.
Divide both sides by $-7$ and reverse the sign.
$x >$ q
Dividing by a negative number turns $<$ into $>$.
Choose a test value: the boundary plus one.
$x =$ t
A number just past the boundary, on the shaded side, should make the inequality true.
Substitute the test value in the original inequality.
$26 - 7 \times (\text{test value}) < -30 \text{ is true}$
The original inequality holds, so the reversed sign was right.
Solve $4x + 15 < -5$. What does $4x$ have to be less than, and what does $x$ have to be less than?
$4x <$ p, so $x <$ n
Solve $-6x + 13 \ge 55$. Give the solution set of $x$.
This task has no paper form; do it on a device.
Solve $6x - 11 > 37$. Give the solution set of $x$.
This task has no paper form; do it on a device.
Ana has saved $19$ dollars toward a bike that costs $102$ dollars. She saves $11$ dollars every week. How much more does she need, and what is the fewest whole weeks she must save to have at least $102$ dollars?
Still needed: d dollars. Fewest weeks: n
Diego has a gift card worth $40.00$ dollars for an online craft store. Every order pays a $8$-dollar shipping fee, and each pack of beads costs $6$ dollars. How much of the card can go on beads, and what is the most packs he can order in one order?
For beads: b dollars. Most packs: n
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
An airline charges extra for a checked bag over 50 pounds. Kai's packed suitcase weighs $33.75$ pounds, and he wants to add some books that weigh $2.25$ pounds each. How many pounds of room are left, and what is the most books he can add without going over the limit?
Room left: r lb. Most books: n
You can solve and graph a two-step inequality. Without looking: what happens to the inequality sign when you divide both sides by $-2$, and why? Solve $-5x + 2 > 17$ and say which way to shade.
19. Your turn: solve $\dfrac{x}{4} - 3 \le 2$, step 2
$x \le 20$
4 is positive, so the sign stays the same.
19. Your turn: solve $\dfrac{x}{4} - 3 \le 2$, step 3
$\text{filled circle at } 20, \text{ shaded left}; \quad x = 0: -3 \le 2 \text{ true}$
The sign $\le$ includes 20, and a value from the shaded side works.