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Solve and graph linear inequalities in one variable, and handle compound inequalities and absolute value equations and inequalities.
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 learn that an inequality is solved with the same moves as an equation, with one extra rule: multiplying or dividing both sides by a negative number turns the sign round. You then read the answer as a set of numbers on a number line. The second half joins conditions with 'and' and 'or', and shows that $|x - a|$ is simply the distance from $x$ to $a$ — which makes absolute value equations and inequalities questions about a number line rather than about bars.
You already solve $4x - 3 = 9$, and you already read a number line from left to right. An inequality is solved with the very same moves; the only new rule is one thing that happens when a negative number gets involved, and the only new output is a range of answers rather than a single one.
Inequality: a statement with $<$, $\le$, $>$ or $\ge$ instead of $=$.
Solution set: every number that makes the statement true — usually a ray or an interval, not one value.
Strict: $<$ and $>$, which exclude the endpoint. Weak: $\le$ and $\ge$, which include it.
At most / at least: 'at most $40$' is $\le 40$; 'at least $40$' is $\ge 40$.
Every move that is legal on an equation is legal here — add, subtract, multiply or divide both sides by the same thing — with one extra rule: multiplying or dividing both sides by a negative number reverses the direction of the sign. That is not a convention; it is what negation does to order, since $2 < 5$ but $-2 > -5$. So $-2x + 1 \le 7$ becomes $-2x \le 6$ and then $x \ge -3$. The answer is a set of numbers, drawn on a number line with a closed circle when the endpoint is included and an open one when it is not, shaded towards the solutions. In a real situation the words fix the sign: 'at most' is $\le$, 'at least' is $\ge$, and the answer is often rounded to a whole number of things.
Another way: diagram
A number line from $-6$ to $6$ with a closed circle at $-3$ and the ray to the right shaded, labelled $x \ge -3$; beneath it the same line with an open circle at $3$ and the ray to the right shaded, labelled $x > 3$.
Another way: story
Picture the number line as a see-saw. Adding the same weight to both sides leaves the tilt alone; turning the whole line back to front — which is what multiplying by a negative does — swaps which side is higher, so the sign has to turn with it.
"Flip the sign whenever there is a minus." Only when you multiply or divide both sides by a negative number. Subtracting $5$ from both sides changes nothing about the direction.
"$x > 3$ means $x = 4$." It means every number above $3$ — $3.5$, $\pi$, $1000$. Whole numbers only appear when the situation says so, as with a number of visits.
"An open and a closed circle are the same picture." They are the answer to 'is the endpoint itself a solution?'. $x \ge 2$ includes $2$; $x > 2$ does not.
Add $3$ to both sides: $4x > 12$.
Adding never changes the direction.
Divide both sides by $4$, a positive number: $x > 3$.
Positive division keeps the sign.
On a number line: an open circle at $3$, shaded to the right.
Strict, so the endpoint is excluded.
Subtract $1$: $-2x \le 6$.
Still no change of direction.
Divide by $-2$: the sign turns, giving $x \ge -3$.
Dividing by a negative reverses it.
Check with $x = 0$: $-2(0) + 1 = 1$, and $1 \le 7$ is true, so $0$ should be in the set — and it is.
Subtract $2x$ and then $2$ from both sides: $3x < 9$.
Divide by $3$, which is positive, so the sign stays: $x < 3$.
Solve $7x - 8 > 48$. Fill in the number.
$x > $ n
Solve $-5x - 9 \le -19$. Fill in the number.
$x \ge $ n
Solve $5x + 7 < x - 5$. Fill in the number.
$x < $ n
How is $x \ge 6$ drawn on a number line?
A player has scored $62$ points so far this season and scores exactly $10$ points in every remaining game. The club record is $131$ points. What is the smallest number of further games after which the player passes the record?
Answer:
You already know that $|-4| = 4$, and you have just solved simple inequalities. What is new is joining two conditions with 'and' or 'or', and reading $|x - a|$ as a distance — which turns nearly every absolute value question into a sentence about a number line.
Compound inequality: two conditions joined. 'And' keeps only what satisfies both; 'or' keeps anything that satisfies either.
$|x - a|$: the distance between $x$ and $a$ on the number line. It is never negative.
Interval: a band of numbers, such as $-2 < x \le 3$.
Tolerance: how far from a target value a measurement may be, written $|m - \text{target}| \le \text{tolerance}$.
Two conditions can be joined in two ways. And means both hold, so the solution is the overlap, and it is often written as a band: $-3 < 2x + 1 \le 7$ is solved by doing the same thing to all three parts. Or means either holds, so the solution is the two pieces together, such as $x < 1$ or $x > 4$. Absolute value connects the two, because $|x - a|$ is the distance from $x$ to $a$: $|x - a| = d$ says $x$ is exactly $d$ away, which happens at the two points $a - d$ and $a + d$; $|x - a| < d$ says $x$ is nearer than $d$, an and — the band $a - d < x < a + d$; and $|x - a| > d$ says $x$ is further than $d$, an or — the two rays outside. Because a distance is never negative, $|x - a| = -d$ with $d$ positive has no solution at all.
Another way: diagram
A number line with $4$ marked in the middle and arrows of length $6$ drawn to $-2$ and to $10$, labelled 'distance $6$ either way', showing the two solutions of $|x - 4| = 6$.
Another way: story
Stop reading the bars as an operation and start reading them as 'how far apart'. 'Within $2$ mm of $50$ mm' is $|m - 50| \le 2$, and the answer, $48$ to $52$, is obvious before any algebra.
"$|x| = -3$ has a solution somewhere." It has none. An absolute value is a distance and cannot be negative, so no $x$ works.
"$|x - 4| = 6$ means $x - 4 = 6$." That is only one of the two cases. The other is $x - 4 = -6$, giving $x = -2$ as well as $x = 10$.
"'And' and 'or' are much the same." $x < 1$ and $x > 4$ describes no numbers at all; $x < 1$ or $x > 4$ describes almost all of them. Which word the problem uses decides the answer.
Subtract $1$ from all three parts: $-4 < 2x \le 6$.
Every move applies to all three.
Divide all three by $2$, a positive number: $-2 < x \le 3$.
Both signs keep their direction.
So the solutions are the numbers above $-2$ and up to and including $3$.
Read it as a distance: $x$ is exactly $6$ away from $4$.
The bars mean 'how far from'.
Six to the left is $4 - 6 = -2$; six to the right is $4 + 6 = 10$.
Two cases, one each side.
So $x = -2$ or $x = 10$.
$|x|$ is the distance from $x$ to $0$, so $x$ is nearer than $3$ to zero.
That is the band $-3 < x < 3$.
Solve $8 < 6x - 4 \le 14$. Fill in both numbers.
p $ < x \le $ q
Solve $|x - 8| = 7$. Fill in the solutions, smaller first.
$x = $ a $\ $ or $\ x = $ b
Solve $|x + 1| < 8$. Fill in both ends.
a $ < x < $ b
How many solutions does $|x + 9| = -8$ have?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A gym charges $\$40$ to join and $\$7$ per visit. With $\$68$ to spend, what is the greatest number of visits you can make?
answer
A part must measure $45$ mm with a tolerance of $5$ mm, written $|m - 45| \le 5$. What is the largest acceptable $m$?
Answer:
You can solve and graph linear inequalities, compound inequalities and absolute value statements. Explain why solving $-2x \le 6$ gives $x \ge -3$ and not $x \le -3$, and why $|x + 1| = -2$ has no solution.
8. Your turn: solve $5x + 2 < 2x + 11$, step 2
20. Your turn: solve $|x| < 3$, step 2