Solve multi-step inequalities using the same techniques as equations, with the critical addition of reversing the inequality when multiplying or dividing by a negative. Graph solution sets on a number line and write them in interval notation.
A lesson of Algebra 1, in Linear Inequalities & Absolute Value, about 35 minutes.
By the end of this lesson, you will be able to solve multi-step inequalities including the flip rule for negatives, graph a solution set on a number line with open and closed circles, and write it in interval notation.
An Equation Has an Answer. An Inequality Has a Range.
$$x + 3 = 7$$
That has exactly one solution, $x = 4$. Nothing else works.
$$x + 3 < 7$$
That has infinitely many. Every number below $4$ works: $3$, $0$, $-11$,
$3.999$. The solution is not a number, it is a set of numbers, and you
describe it rather than list it.
$$x < 4$$
Inequalities are how constraints get written down. Budgets, speed limits,
weight limits, age restrictions and passing grades are all "at most" or "at
least" statements, and none of them can be written as an equation.
The good news. Solving one works exactly like solving an equation, with
exactly one extra rule. That rule is the whole difficulty of this lesson.
The Flip Rule
When you multiply or divide both sides of an inequality by a negative
number, the inequality sign turns round.
Here is why, in one line you can check yourself. Start with something true:
$$2 < 8$$
Multiply both sides by $-1$:
$$-2 \quad \text{and} \quad -8$$
Now $-2$ is the bigger of the two, because it is closer to zero. So the
true statement is
$$-2 > -8$$
The sign turned round because multiplying by a negative reflects both numbers
to the other side of zero, and reflecting reverses which one is larger. It is
not a convention someone chose. Leave the sign as it was and the statement
becomes false.
Adding or subtracting a negative does not trigger it. Only multiplying or
dividing by one does.
$$x - (-3) > 5 \quad \text{is} \quad x + 3 > 5 \quad \text{so} \quad x > 2$$
No flip there. There is a negative in sight, but nothing was multiplied by it.
Writing the Answer Down
On a number line:
Open circle for $<$ and $>$. The endpoint is not a solution
Closed circle for $\leq$ and $\geq$. The endpoint is a solution
A round bracket excludes the endpoint, a square bracket includes it. Infinity
always takes a round bracket, because you never arrive there.
Example 1: No Negatives, No Flip
Here is the whole method on one inequality, worked all the way through.
Solve $3x - 5 > 7$.
Add 5 to both sides and simplify 3x > 12
Divide both sides by 3 and simplify x > 4
Write the solution in interval notation (4, \infty)
Test it. Put x = 6 into the left hand side of the original 13
Worked Example 1: Your Turn, Still No Flip
Solve $2x + 3 < 11$.
Example 2: Dividing by a Negative
Solve $-4x + 6 \geq 22$.
Watch the last step. This is the one that catches people.
Subtract 6 from both sides and simplify -4x >= 16
Divide both sides by -4, and turn the sign round x <= -4
Write the solution in interval notation (-\infty, -4]
Test it. Put x = -6 into the left hand side of the original 30
Worked Example 2: Flip It Yourself
Solve $-3x + 1 > 10$.
Example 3: Several Steps, Variables Both Sides
Everything you know about linear equations still applies. Distribute, collect,
then finish.
Solve $2(3x - 1) < 4x + 6$.
Multiply out the bracket 6x - 2 < 4x + 6
Subtract 4x from both sides and simplify 2x - 2 < 6
Add 2 to both sides and simplify 2x < 8
Divide both sides by 2 and simplify x < 4
Graph It: The Solution Set
Three decisions each time: where the endpoint goes, whether it is open or
closed, and which way the shading runs.
Worked Example 3: Brackets and Both Sides
Solve $5(x + 2) \geq 3x + 16$.
Worked Example 4: A Budget
You have at most $$200$ to spend. There is already $$35$ of shopping in
your basket, and every extra item costs $$15$.
"At most" is the phrase that makes this an inequality. Let $n$ be the number of
extra items.
Example 4: What the Answer Actually Says
An inequality's solution is a set, and it is worth being able to read one out
loud.
$$x \leq -4$$
"Every number that is $-4$ or below." That includes $-4$ itself, $-4.5$, $-100$,
and nothing above $-4$.
On a number line: a filled circle sits at $-4$ and the shading runs left
forever. The filled circle is doing real work. It is the difference between
$-4$ being a solution and not being one, which for a budget or a weight limit
is the difference between allowed and not allowed.
In interval notation:
$$(-\infty, -4]$$
Read that from left to right as "from negative infinity, up to and including
$-4$". The square bracket is the closed circle, written down. The round bracket
next to infinity is there because infinity is a direction, not a number you can
include.
Three notations, one set of numbers. Which one you use depends on who is
reading, and being able to move between them is most of what this lesson is
for.
Common Mistakes
Mistake 1: Forgetting to flip
Wrong: $-3x > 12$ gives $x > -4$
Right: $-3x > 12$ gives $x < -4$
This is the most common error in the topic by a wide margin. Every time you
divide or multiply by a negative, stop and ask whether you turned the sign
round.
Mistake 2: Flipping when you should not
Wrong: $x - (-3) > 5$, "there is a negative, so flip"
Right: $x + 3 > 5$, so $x > 2$, no flip
The rule is about multiplying and dividing. A negative sitting in an addition
does not trigger it.
Mistake 3: The wrong kind of circle
$x < 4$ takes an open circle at $4$, because $4$ is not a solution.
$x \leq 4$ takes a filled one. In a real constraint that dot is the difference
between a limit you may reach and one you may not.
Mistake 4: A square bracket next to infinity
Wrong: $(-\infty, 4]$ written as $[-\infty, 4]$
Infinity is not a number, so it can never be included.
Where This Goes Next
There is a way to never flip anything.
$$-3x > 12$$
Instead of dividing by $-3$, add $3x$ to both sides and subtract $12$:
$$-12 > 3x$$
$$-4 > x$$
which says the same thing as $x < -4$. Every step was an addition, so the flip
rule never came up.
That is not a trick to replace the rule with, because reading $-4 > x$
backwards has its own trap. It is worth knowing because it shows what the flip
rule really is: a shortcut for moving a term across, and not a special
exception living inside inequalities.
Next comes what happens when two inequalities apply at once, which is how most
real constraints arrive. A temperature must be above freezing and below
boiling. A dose must be at least effective and at most safe.
Check Your Understanding
Five of them, no instructions this time.
Summary
Same as an equation, plus one rule:
What you do
The sign
Add or subtract anything
Stays
Multiply or divide by a positive
Stays
Multiply or divide by a negative
Turns round
Graphing: open circle for $<$ and $>$, filled circle for $\leq$ and
$\geq$, shading toward the solutions.
Interval notation: round bracket excludes, square bracket includes, and
infinity is always round.
Check with two test values, one inside your solution set and one outside.
Both have to behave, and that catches a missed flip immediately.