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Free guide covering 30+ worked examples across every major inequality type

How to Solve Inequalities

Q: What is the solution set of an inequality?

The solution set is the collection of all values that make the inequality true. It can be written with inequality notation, interval notation, set-builder notation, or a number-line graph.

Q: When do you flip the inequality sign?

You flip the inequality sign when you multiply or divide both sides by a negative number, or when you rewrite the comparison by swapping the sides.

Q: Why does the inequality sign flip when you multiply by a negative number?

Multiplying by a negative reverses order on the number line. Since 2 -5, the comparison must flip to stay true.

A complete step-by-step guide covering all 6 types of inequalities — from basic linear to quadratic and rational. Learn the key rules, see worked examples, graph solutions on a number line, and write answers in interval notation. Free, no calculator required.

solving inequalities step by stepwhen to flip inequality signinequality solution set interval notation

What this guide covers

The core inequality rules, including the exact moments when the comparison sign must flip.

Seven solving workflows: linear, multi-step, compound, fractions, absolute value, quadratic, and rational.

Number-line graphing, interval notation, practice problems, FAQ, and direct links into the matching calculators.

📐

Linear Inequalities

x + 3 > 7

Jump to section ↓

🔢

Multi-Step

2x - 1 <= 5

Jump to section ↓

🔀

Compound AND / OR

-1 < x <= 5

Jump to section ↓

📊

Absolute Value

|x - 2| < 5

Jump to section ↓

📈

Quadratic Inequalities

x^2 - 5x + 6 < 0

Jump to section ↓

➗

Rational Inequalities

(x+1)/(x-2) > 0

Jump to section ↓

Linear Multi-Step Compound Fractions Absolute Value Quadratic Rational FAQ

Zone 2

What Is an Inequality? Key Symbols and Definitions

An inequality compares two expressions without requiring them to be exactly equal. Instead of finding one exact value, you usually find a set of values that make the statement true.

That set can be a ray, a bounded interval, a union of two intervals, all real numbers, or no real solution. The correct output depends on both the algebra and the meaning of the symbol you started with.

In practice, solving inequalities means preserving a true order statement while simplifying the algebra. The main difference from equations is that some operations, especially multiplying or dividing by a negative number, change the direction of the comparison.

Symbol	Meaning	Example	Number Line	Endpoint Rule
>	Greater than	x > 3	--o====>	Open endpoint because 3 is not included.
<	Less than	x < -2	<====o--	Open endpoint because -2 is not included.
>=	Greater than or equal to	x >= 5	--●====>	Closed endpoint because 5 is included.
<=	Less than or equal to	x <= 0	<====●--	Closed endpoint because 0 is included.
!=	Not equal to	x != 2	<====o--o====>	The graph excludes one point and keeps both outer rays.

Equation

2x + 3 = 7 \Rightarrow x = 2

An equation usually gives one exact value or a finite set of exact values.

Inequality

2x + 3 > 7 \Rightarrow x > 2

An inequality usually gives a region of values, not a single point.

The Golden Rules of Solving Inequalities

Add or subtract the same value on both sides

The inequality sign stays the same.

x + 5 > 8 -> x > 3

Multiply or divide by a positive number

The inequality sign stays the same.

3x < 12 -> x < 4

Multiply or divide by a negative number

The inequality sign flips.

-3x < 12 -> x > -4

Swap the left and right sides

The inequality sign flips.

5 > x -> x < 5

When Do You Flip the Inequality Sign?

You flip the inequality sign when an operation reverses the order of numbers on the number line. Multiplying or dividing by a negative number does exactly that, so the comparison must reverse to stay true.

A quick mental check is to compare two simple numbers. Since 2 < 5, multiplying both by -1 gives -2 > -5. The direction changed because the number line order reversed.

You do not flip the sign when adding, subtracting, multiplying by a positive number, or dividing by a positive number. Those operations preserve order.

Flip The Sign

Multiply both sides by a negative number

-2x > 6 -> x < -3

Divide both sides by a negative number

-x/3 <= 4 -> x >= -12

Rewrite the comparison after swapping sides

7 < x -> x > 7

Do Not Flip

Add or subtract any number

x + 5 > 3 -> x > -2

Multiply or divide by a positive number

2x <= 10 -> x <= 5

Number line intuition

Start with 2 < 5. Multiplying by -1 gives -2 > -5. The direction flips because the negative sign reverses left and right order on the number line.

How to Solve Linear Inequalities Step by Step

A linear inequality has degree 1, so the variable never appears in a power higher than one. These are the foundation for every other inequality type because the goal is to isolate the variable while keeping the comparison true.

Most linear problems produce one half-infinite interval such as x > 4 or x <= -3. The main danger is forgetting that the sign flips only when you divide or multiply by a negative number.

Simplify both sides and combine like terms.

Move variable terms to one side and constants to the other.

Isolate the variable by dividing by its coefficient.

If the coefficient is negative, flip the sign before writing the final answer.

Translate the answer into interval notation and a number-line graph.

Example 1

x + 3 > 7

Subtract 3 from both sides: x + 3 - 3 > 7 - 3.

Simplify to get x > 4.

Because the inequality is strict, 4 is not included.

Solution

x > 4

Interval

(4, \infty)

Number Line

--o========>

Use an open endpoint because the symbol is >.

Example 2

-3x \ge 12

Divide both sides by -3.

Because you divided by a negative number, flip the sign.

The result is x <= -4.

Solution

x \le -4

Interval

(-\infty, -4]

Number Line

<========●--

This is the standard flip-sign example.

Example 3

5 > 2x - 1

Add 1 to both sides to get 6 > 2x.

Divide by 2 to get 3 > x.

Rewrite in standard form as x < 3.

Solution

x < 3

Interval

(-\infty, 3)

Number Line

<========o--

Swapping sides also reverses the visible direction of the comparison.

Try it yourself

Linear Inequality Calculator

Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.

Open Full Linear Inequality Calculator

Example input: 2x + 5 > 11

How to Solve Multi-Step Inequalities

Multi-step inequalities are still linear in spirit, but they contain several algebra moves: distribution, combining like terms, moving variables on both sides, or clearing grouped expressions.

The safest way to solve them is to treat the problem like a checklist. Distribute first, simplify second, isolate the variable third, and only then decide whether the final division causes a sign flip.

Distribute through parentheses and clear grouped expressions.

Combine like terms on each side before moving anything across the sign.

Move variable terms to one side and constants to the other.

Divide by the final coefficient, flipping the sign only if that coefficient is negative.

Check the final inequality against the original structure.

Example 1

3(x - 2) + 4 > 2x + 1

Distribute to get 3x - 6 + 4 > 2x + 1.

Combine like terms: 3x - 2 > 2x + 1.

Subtract 2x, then add 2, to get x > 3.

Solution

x > 3

Interval

(3, \infty)

Number Line

--o========>

Distribution creates the clean linear form.

Example 2

5 - 2x \le 3x + 15

Subtract 3x from both sides to get 5 - 5x <= 15.

Subtract 5 to get -5x <= 10.

Divide by -5 and flip the sign: x >= -2.

Solution

x \ge -2

Interval

[-2, \infty)

Number Line

--●========>

This is the classic variables-on-both-sides case with a flip.

Example 3

2(3x + 1) \ge 4(x - 3) + 6

Distribute to get 6x + 2 >= 4x - 12 + 6.

Combine like terms: 6x + 2 >= 4x - 6.

Subtract 4x and then subtract 2 to get 2x >= -8, so x >= -4.

Solution

x \ge -4

Interval

[-4, \infty)

Number Line

--●========>

Because the last division is by +2, the sign stays the same.

Try it yourself

Linear Inequality Calculator

Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.

Solve a Multi-Step Linear Inequality

Example input: 3(2x-1)+4>5x+8

How to Solve Compound Inequalities (AND / OR)

Compound inequalities combine two comparisons into one answer. The key idea is logical structure: AND means both conditions must be true at the same time, while OR means at least one branch can be true.

That logic determines the solution form. AND answers usually become one bounded interval, while OR answers become a union of two separate intervals.

Decide whether the problem is AND or OR.

For three-part chains, perform the same operation on all parts at once.

For split comparisons, solve each branch separately.

Take the intersection for AND and the union for OR.

Match endpoints with the original strict or inclusive symbols.

Example 1

-1 < 2x + 3 \le 9

Subtract 3 from all three parts to get -4 < 2x <= 6.

Divide all parts by 2. Because 2 is positive, the signs stay the same.

The result is -2 < x <= 3.

Solution

-2 < x \le 3

Interval

(-2, 3]

Number Line

--o======●--

This is an AND chain, so the answer is one interval.

Example 2

-6 \le -2x + 2 < 4

Subtract 2 from all parts: -8 <= -2x < 2.

Divide all parts by -2, and flip every comparison sign.

Rewrite 4 >= x > -1 as -1 < x <= 4.

Solution

-1 < x \le 4

Interval

(-1, 4]

Number Line

--o======●--

When the shared coefficient is negative, the whole chain flips.

Example 3

x < -3 \; \text{or} \; x > 7

Solve each branch independently. No extra algebra is needed here.

The left branch gives (-\infty, -3).

The right branch gives (7, \infty), and OR combines them with a union.

Solution

x < -3 \; \text{or} \; x > 7

Interval

(-\infty, -3) \cup (7, \infty)

Number Line

<====o------o====>

OR keeps both valid outer rays.

Try it yourself

Compound Inequality Calculator

Solve compound inequalities with interval intersection, union logic, and graph output.

Open Full Compound Inequality Calculator

Example input: -3 < 2x + 1 <= 7

How to Solve Inequalities with Fractions

An inequality with fractions is still usually linear as long as the denominator does not contain the variable. The cleanest strategy is to multiply every term by the least common denominator so the fractions disappear at once.

This is different from rational inequalities. Here the denominator is just a number, so multiplying through by the LCD does not create sign-chart issues.

Identify the least common denominator of all fractional terms.

Multiply every term by that LCD.

Simplify the resulting linear inequality.

If you multiply or divide by a negative value, flip the sign.

Write the final interval and graph.

Example 1

\frac{x}{3} + \frac{1}{2} > \frac{5}{6}

The LCD is 6, so multiply every term by 6.

That gives 2x + 3 > 5.

Subtract 3 and divide by 2 to get x > 1.

Solution

x > 1

Interval

(1, \infty)

Number Line

--o========>

Clearing the fractions first keeps the algebra tidy.

Example 2

-\frac{x}{4} \le 3

Multiply both sides by -4.

Because -4 is negative, flip the sign.

The result is x >= -12.

Solution

x \ge -12

Interval

[-12, \infty)

Number Line

--●========>

A negative fractional coefficient still triggers the flip rule.

Example 3

\frac{2x-1}{3} \ge \frac{x+2}{4} - 1

Use the LCD 12 to clear fractions: 4(2x-1) >= 3(x+2) - 12.

Expand to get 8x - 4 >= 3x - 6.

Subtract 3x and add 4 to get 5x >= -2, so x >= -2/5.

Solution

x \ge -\frac{2}{5}

Interval

\left[-\frac{2}{5}, \infty\right)

Number Line

--●========>

Fractions in the coefficients do not change the rule set; they just add arithmetic.

Try it yourself

Linear Inequality Calculator

Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.

Practice Fractions in the Linear Calculator

Example input: x/3 + 1/2 > 5/6

How to Solve Absolute Value Inequalities

Absolute value measures distance from zero or from a center point. That makes the symbol interpretation decisive: less-than cases stay inside a region, and greater-than cases move to the outside.

The fastest way to solve absolute value inequalities is to translate them into compound linear inequalities. Once the absolute value bars disappear, the problem becomes a familiar linear or compound task.

Isolate the absolute value expression first.

If the form is |A| < k or |A| <= k, rewrite it as -k < A < k or -k <= A <= k.

If the form is |A| > k or |A| >= k, rewrite it as A < -k or A > k, or with inclusive endpoints if needed.

Solve the resulting linear inequalities.

Check for special cases such as |A| < 0 or |A| >= 0.

Pattern	Rewrite Rule	Meaning
\|A\| < k or \|A\| <= k	Rewrite as -k < A < k or -k <= A <= k.	The value stays inside a distance band.
\|A\| > k or \|A\| >= k	Rewrite as A < -k or A > k, with inclusive endpoints when needed.	The value stays outside the central band.
\|A\| < 0	No solution.	Absolute value is always >= 0.
\|A\| >= 0	All real numbers.	Every absolute value is nonnegative.

Example 1

|x - 2| < 5

Use the inside rule: -5 < x - 2 < 5.

Add 2 to every part to get -3 < x < 7.

Because the original inequality was strict, both endpoints stay open.

Solution

-3 < x < 7

Interval

(-3, 7)

Number Line

--o======o--

Less-than cases create one bounded interval.

Example 2

|2x + 1| \ge 7

Use the outside rule: 2x + 1 <= -7 or 2x + 1 >= 7.

Solve the left branch to get x <= -4.

Solve the right branch to get x >= 3, then combine them with a union.

Solution

x \le -4 \; \text{or} \; x \ge 3

Interval

(-\infty, -4] \cup [3, \infty)

Number Line

<====●------●====>

Greater-than cases produce two outer branches.

Example 3

|3x - 1| < -2

Absolute value is never negative.

That means |3x - 1| cannot be less than -2.

So the inequality has no real solution.

Solution

\varnothing

Interval

\varnothing

Number Line

(empty)

Always check the right-hand side before splitting the inequality.

Try it yourself

Absolute Value Inequality Calculator

Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.

Open Full Absolute Value Inequality Calculator

Example input: |x - 2| <= 4

How to Solve Quadratic Inequalities

Quadratic inequalities are solved with interval testing. First find the real roots of the related equation, then use those critical points to divide the number line into sign regions.

The graph of the parabola explains why answers come in patterns. If the parabola opens upward, it is usually positive outside the roots and negative between them. If it opens downward, the pattern reverses.

Move all terms to one side so the other side is zero.

Factor or use the quadratic formula to find the real roots.

Plot the roots on the number line.

Test one point from each interval to determine the sign.

Keep the intervals that satisfy the original inequality and include endpoints only for <= or >=.

a > 0 and q(x) > 0

Outside the two roots, if two real roots exist.

a > 0 and q(x) < 0

Between the two roots, if two real roots exist.

a > 0 and discriminant < 0

Always positive, so > 0 gives all real numbers.

a < 0

Reverse the inside-outside pattern because the parabola opens downward.

Example 1

x^2 - 5x + 6 < 0

Factor: x^2 - 5x + 6 = (x-2)(x-3).

The roots are 2 and 3, so test the intervals (-\infty, 2), (2, 3), and (3, \infty).

The product is negative only between the roots, so the answer is 2 < x < 3.

Solution

2 < x < 3

Interval

(2, 3)

Number Line

--o==o--

For an upward parabola, the inside interval is where the quadratic is negative.

Example 2

x^2 - x - 6 \ge 0

Factor: x^2 - x - 6 = (x-3)(x+2).

The roots are -2 and 3.

Testing intervals shows the quadratic is nonnegative outside the roots, and >= includes the roots.

Solution

x \le -2 \; \text{or} \; x \ge 3

Interval

(-\infty, -2] \cup [3, \infty)

Number Line

<====●------●====>

Outside intervals are common when the quadratic must be positive.

Example 3

x^2 + 2x + 5 > 0

Compute the discriminant: b^2 - 4ac = 4 - 20 = -16.

Because the discriminant is negative, there are no real roots.

The leading coefficient is positive, so the parabola stays above the x-axis for all real x.

Solution

x \in \mathbb{R}

Interval

(-\infty, \infty)

Number Line

<============>

No real roots does not mean no solution; it depends on the direction of the parabola.

Try it yourself

Quadratic Inequality Calculator

Solve quadratic inequalities with sign analysis, roots, interval notation, and a number-line graph.

Open Full Quadratic Inequality Calculator

Example input: x^2 - 7x + 10 <= 0

How to Solve Rational Inequalities

A rational inequality has a variable in the denominator, so you cannot safely multiply both sides by that denominator without knowing its sign. Instead, solve the problem with a sign chart.

The sign-chart method tracks both zeros of the numerator and undefined points from the denominator. Those critical points divide the number line into regions where the sign of the rational expression is stable.

Move all terms to one side and combine them into one fraction.

Find the zeros of the numerator and the values that make the denominator zero.

Plot all critical points on the number line.

Test one value in each interval to determine the sign of the full rational expression.

Include zeros only when the inequality allows equality, and never include denominator zeros.

Example 1

\frac{x+1}{x-2} > 0

The numerator is zero at x = -1 and the denominator is zero at x = 2.

These points create the intervals (-\infty, -1), (-1, 2), and (2, \infty).

The fraction is positive in the first and third intervals, so the answer is (-\infty, -1) union (2, \infty).

Solution

x < -1 \; \text{or} \; x > 2

Interval

(-\infty, -1) \cup (2, \infty)

Number Line

<====o------o====>

Both critical points are excluded here.

Example 2

\frac{x-3}{x+1} \le 0

The numerator is zero at x = 3 and the denominator is zero at x = -1.

Testing intervals shows the expression is negative on (-1, 3).

Because <= allows zero, include x = 3, but never include x = -1.

Solution

-1 < x \le 3

Interval

(-1, 3]

Number Line

--o======●--

Undefined points are always excluded, even for <= or >=.

Example 3

\frac{2}{x-1} > 1

Move everything to the left: 2/(x-1) - 1 > 0.

Combine into one fraction: (3-x)/(x-1) > 0.

The critical points are x = 1 and x = 3, and the positive interval is (1, 3).

Solution

1 < x < 3

Interval

(1, 3)

Number Line

--o==o--

This is why you should rearrange first instead of cross-multiplying blindly.

Try it yourself

Rational Inequality Calculator

Solve rational inequalities with steps, excluded values, sign charts, and interval notation.

Open Full Rational Inequality Calculator

Example input: (x+1)/(x-2) > 0

How to Graph Inequalities on a Number Line

A number-line graph turns an algebraic answer into a visible region. First mark the boundary values, then choose open or closed endpoints, and finally shade the direction or interval that satisfies the inequality.

Use an open endpoint for strict inequalities such as < and >. Use a closed endpoint for inclusive inequalities such as <= and >=. For split answers, draw two separate rays instead of forcing the graph into one segment.

Graphing is one of the fastest ways to catch a wrong answer. If your interval notation says one ray but your graph looks bounded, go back and check the endpoint logic or the sign chart.

Open endpoint

Use it for < or > because the boundary value is excluded.

--o====>

Closed endpoint

Use it for <= or >= because the boundary value is included.

--●====>

Split graph

Use two rays when the answer is a union, not one continuous segment.

<====o--o====>

How to Write the Solution in Interval Notation

Interval notation compresses the final solution into a standard set language. Parentheses mean the endpoint is excluded, while brackets mean the endpoint is included.

Infinity always uses parentheses because infinity is not a reachable real endpoint. So x > 4 becomes (4, infinity) and x <= -2 becomes (-infinity, -2].

Union symbols connect separated solution sets. That is common in OR compound inequalities, absolute value outside cases, quadratic inequalities that are positive outside the roots, and many rational inequalities.

Solution	Interval Notation	Graph Reading
$x > 4$	$(4, \infty)$	open at 4, shade right
$x \le -2$	$(-\infty, -2]$	closed at -2, shade left
$-1 < x \le 5$	$(-1, 5]$	open at -1, closed at 5
$x < -3 \; \text{or} \; x > 7$	$(-\infty, -3) \cup (7, \infty)$	two separate rays

Open the Interval Notation Calculator

Zone 4

Inequality Type Comparison Chart

Type	Typical Form	Key Rule	Solution Form
Linear	ax + b > c	Flip only if you divide or multiply by a negative.	One ray or one interval boundary.
Multi-Step	a(bx+c) >= d	Distribute and combine like terms before isolating x.	Usually one ray.
Compound AND	a < bx + c <= d	Operate on all parts together or intersect separate branches.	One bounded interval.
Compound OR	bx + c < a or bx + c > d	Solve each branch separately and take the union.	Two rays.
Fractions	x/a + b > c	Clear denominators with the LCD.	Usually one ray.
Absolute Value	\|ax+b\| < k or \|ax+b\| > k	Inside rule for <, outside rule for >.	One interval or a union of two rays.
Quadratic	ax^2 + bx + c > 0	Find roots, then test intervals.	One interval, two intervals, all reals, or empty.
Rational	f(x)/g(x) > 0	Use a sign chart. Never include denominator zeros.	One or more intervals.

Zone 5

Solving Inequalities Practice Problems

These ten practice problems bring the worked-example total on this page above thirty. Try each one first, then open the answer to compare the logic, interval notation, and number-line graph.

Practice Problem 1: Linear (Basic)

4x - 7 > 9

Add 7 to both sides: 4x > 16.

Divide by 4: x > 4.

Solution

x > 4

Interval

(4, \infty)

Number Line

--o========>

Practice Problem 2: Linear (flip) (Basic)

-5x + 3 \le 18

Subtract 3: -5x <= 15.

Divide by -5 and flip the sign: x >= -3.

Solution

x \ge -3

Interval

[-3, \infty)

Number Line

--●========>

Practice Problem 3: Multi-Step (Intermediate)

3(2x-1) + 4 > 5x + 8

Distribute: 6x - 3 + 4 > 5x + 8, so 6x + 1 > 5x + 8.

Subtract 5x: x + 1 > 8.

Subtract 1: x > 7.

Solution

x > 7

Interval

(7, \infty)

Number Line

--o========>

Practice Problem 4: Compound AND (Intermediate)

-3 \le 2x + 1 < 7

Subtract 1 from all parts: -4 <= 2x < 6.

Divide all parts by 2: -2 <= x < 3.

Solution

-2 \le x < 3

Interval

[-2, 3)

Number Line

--●======o--

Practice Problem 5: Fractions (Intermediate)

\frac{x}{2} - \frac{1}{3} \ge \frac{5}{6}

Multiply every term by 6: 3x - 2 >= 5.

Add 2: 3x >= 7.

Divide by 3: x >= 7/3.

Solution

x \ge \frac{7}{3}

Interval

\left[\frac{7}{3}, \infty\right)

Number Line

--●========>

Practice Problem 6: Absolute Value <= (Intermediate)

|3x - 6| \le 9

Rewrite as -9 <= 3x - 6 <= 9.

Add 6 to all parts: -3 <= 3x <= 15.

Divide by 3: -1 <= x <= 5.

Solution

-1 \le x \le 5

Interval

[-1, 5]

Number Line

--●======●--

Practice Problem 7: Absolute Value > (Intermediate)

|2x + 5| > 3

Rewrite as 2x + 5 < -3 or 2x + 5 > 3.

Solve to get 2x < -8 or 2x > -2.

Divide by 2: x < -4 or x > -1.

Solution

x < -4 \; \text{or} \; x > -1

Interval

(-\infty, -4) \cup (-1, \infty)

Number Line

<====o--o====>

Practice Problem 8: Quadratic (Advanced)

x^2 - 7x + 10 \le 0

Factor: (x-5)(x-2) <= 0.

The roots are 2 and 5.

For an upward parabola, the expression is nonpositive between the roots, including endpoints.

Solution

2 \le x \le 5

Interval

[2, 5]

Number Line

--●======●--

Practice Problem 9: Rational (Advanced)

\frac{x-1}{x+3} \ge 0

Critical points are x = 1 from the numerator and x = -3 from the denominator.

Testing intervals shows the expression is nonnegative on (-\infty, -3) and [1, \infty).

Include x = 1 because the numerator can be zero. Exclude x = -3 because the denominator cannot be zero.

Solution

x < -3 \; \text{or} \; x \ge 1

Interval

(-\infty, -3) \cup [1, \infty)

Number Line

<====o--●====>

Practice Problem 10: Rational (rearrange) (Advanced)

\frac{3}{x+2} < 1

Move 1 to the left: 3/(x+2) - 1 < 0.

Combine into one fraction: (1-x)/(x+2) < 0.

Critical points are x = 1 and x = -2. Testing intervals gives the solution (-\infty, -2) union (1, \infty).

Solution

x < -2 \; \text{or} \; x > 1

Interval

(-\infty, -2) \cup (1, \infty)

Number Line

<====o--o====>

Zone 6

Frequently Asked Questions

What is an inequality in math?

An inequality is a statement that compares two expressions with symbols such as <, >, <=, or >= instead of saying they are exactly equal.

What is the difference between an equation and an inequality?

An equation usually asks for exact values that make both sides equal. An inequality asks for every value that makes one side larger or smaller than the other, so the answer is often an interval or union of intervals.

What are the four inequality symbols and what do they mean?

The main symbols are > (greater than), < (less than), >= (greater than or equal to), and <= (less than or equal to). Strict symbols exclude the boundary, while inclusive symbols include it.

What is the solution set of an inequality?

The solution set is the collection of all values that make the inequality true. It can be written with inequality notation, interval notation, set-builder notation, or a number-line graph.

When do you flip the inequality sign?

You flip the inequality sign when you multiply or divide both sides by a negative number, or when you rewrite the comparison by swapping the sides.

Why does the inequality sign flip when you multiply by a negative number?

Multiplying by a negative reverses order on the number line. Since 2 < 5 but -2 > -5, the comparison must flip to stay true.

Do you flip the inequality sign when you subtract?

No. Adding or subtracting the same value from both sides does not reverse order, so the inequality sign stays the same.

Do you flip the inequality sign when you divide by a positive number?

No. Dividing by a positive number preserves order, so the sign does not change.

What happens to the inequality sign when you multiply both sides by -1?

It flips. Multiplying both sides by -1 reverses the order of the numbers on the number line.

How do you solve a linear inequality step by step?

Simplify both sides, move variable terms to one side, move constants to the other, isolate the variable, and flip the sign only if you divide or multiply by a negative number.

How do you solve a multi-step inequality?

Distribute first, combine like terms, move the variable terms together, move constants to the other side, and then isolate the variable with the usual sign-flip rule if needed.

How do you solve an inequality with variables on both sides?

Subtract or add variable terms until they are all on one side. Then finish solving like a linear inequality.

How do you solve an inequality with fractions?

Find the least common denominator and multiply every term by it, then solve the resulting linear inequality. If the multiplier is negative, flip the sign.

What is a compound inequality?

A compound inequality combines two inequalities into one statement. It may use AND logic, meaning both conditions must hold, or OR logic, meaning either branch can hold.

How do you solve a compound inequality with AND?

Solve both conditions and keep only the overlap. For three-part chains, perform each operation on all parts at once.

How do you solve a compound inequality with OR?

Solve each branch separately and combine the valid intervals with a union.

What is the difference between AND and OR compound inequalities?

AND keeps the intersection or overlap of the conditions, while OR keeps the union of all branches that work.

How do you solve an absolute value inequality?

Isolate the absolute value first. Then use the inside rule for < or <= and the outside rule for > or >= before solving the resulting linear inequalities.

What is the difference between |x| < k and |x| > k?

|x| < k means x stays within k units of 0, so the answer is a bounded interval. |x| > k means x stays more than k units away from 0, so the answer is two outer rays.

When does an absolute value inequality have no solution?

A problem such as |A| < negative number has no solution because absolute value cannot be negative.

How do you solve a quadratic inequality?

Move everything to one side, find the real roots of the matching quadratic equation, plot them on the number line, test each interval, and keep the intervals that satisfy the original inequality.

What is the sign chart method for solving inequalities?

The sign chart method divides the number line at critical points, then tests each region to determine whether an expression is positive or negative there.

How do you solve a rational inequality?

Move everything to one side, combine into one fraction, find numerator zeros and denominator zeros, make a sign chart, and keep the regions that satisfy the inequality.

Why can't you multiply both sides by a variable when solving rational inequalities?

Because the variable expression could be positive or negative, and that would determine whether the inequality sign should flip. Since you do not know the sign in advance, a sign chart is safer.

How do you write the solution to an inequality in interval notation?

Use parentheses for excluded endpoints, brackets for included endpoints, and unions for split solution sets. Infinity always uses parentheses.

Zone 7

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