Dr. Math Azzam Platform | High School & University Mathematics

Explanation

1. Rules for Inequalities

Addition and subtraction work as they do in equations. The important change happens when multiplying or dividing by a negative number.

A\le B\Rightarrow A+C\le B+C

C>0:\quad A\le B\Rightarrow AC\le BC

C<0:\quad A\le B\Rightarrow AC\ge BC

Common mistakeKey Rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

Worked example

Mini example

-3x\le12

x\ge-4

The sign reverses because both sides were divided by $-3$ .

Final Representation

[-4, ∞)

→

-4

Interactive lab

Linear inequality builder

Divide both sides of

-6x<18

by

-6

.

Which solution follows?

Quick Check

Check the idea before moving on.

What must happen when both sides are divided by a negative number?

-5x>20

Your turn

Solve

-3x+2\le11

.

x\le-3

x\ge-3

x\ge3

x\le3

Explanation

2. Linear & Compound Inequalities

Isolate the variable. In a compound inequality, perform the same operation on all three parts, then write the solution in interval notation.

Worked example

Guided 1 · Linear inequality

3x<9x+4

-6x<4

x>-\frac{2}{3}

The final sign reverses when dividing by $-6$ .

Final Representation

(-2/3, ∞)

→

-2/3

Worked example

Guided 2 · Compound inequality

4\le3x-2<13

6\le3x<15

2\le x<5

[2,5)

Final Representation

[2, 5)

2

5

Interactive lab

Compound inequality builder

Divide all three parts of

-10<-2x\le-4

by

-2

.

Which ordered solution follows?

Interactive lab

Interval and number line builder

Represent the solution you found above. Enter the bounds, choose open or closed endpoints, and shade the matching region.

Represent Your Solution · Required

1 · Build the interval notation.

Left boundary

Right boundary

Interval preview

(, )

2 · Build the matching number line.

?

Quick Check

Check the idea before moving on.

What operation should be performed FIRST on all three parts?

-1<2x-5<7

Explanation

3. Nonlinear Inequalities — Mr. Azzam’s Routine

1. Move all terms to one side and compare with $0$ .
2. Factor completely.
3. Find critical numbers.
4. Divide the number line into intervals.
5. Make a sign chart.
6. Choose the correct interval(s).
7. Check endpoints.

ExplanationRepeated Factors: an even multiplicity factor does not change sign when crossing its zero; an odd multiplicity factor does.

Worked example

Guided 3 · Quadratic inequality

x^2-5x\le-6

x^2-5x+6\le0

(x-2)(x-3)\le0

Critical numbers: $2,3$ . The product is nonpositive between the roots.

Sign chart

Interval	$(-\infty,2)$	$(2,3)$	$(3,\infty)$
Sign	+	−	+
Choose	No	Yes	No

x=2

· included

x=3

· included

[2,3]

Final Representation

[2, 3]

2

3

Worked example

Guided 4 · Repeated factor

x(x-1)^2(x-3)<0

Critical numbers: $0,1,3$ . At $x=1$ the multiplicity is even, so the sign does not change.

Sign by interval: $(-\infty,0):+\quad(0,1):-\quad(1,3):-\quad(3,\infty):+$

Sign chart

Interval	$(-\infty,0)$	$(0,1)$	$(1,3)$	$(3,\infty)$
Sign	+	−	−	+
Choose	No	Yes	Yes	No

x=0

· excluded

x=1

· even multiplicity · excluded

x=3

· excluded

Because the inequality is strict, $x=0,1,3$ are excluded. The repeated zero at $x=1$ does not change the sign, but it still creates a hole in the solution.

(0,1)\cup(1,3)

Final Representation

(0, 1) ∪ (1, 3)

0

3

1

Interactive lab

Polynomial sign chart laboratory

Build the sign chart for

(x-4)(x+1)<0

. Select critical numbers, set signs, and choose the solution intervals.

1 · Select all critical numbers

Zeros divide the real line into intervals. Test one value in each interval.

2 · Cycle each sign: ? → + → −

3 · Select solution intervals

Quick Check

Check the idea before moving on.

List the critical numbers and identify the repeated factor.

(x-4)(x+2)^2<0

Your turn

For

(x-4)(x+2)^2<0

, which solution set correctly treats the repeated zero?

(-\infty,-2)\cup(-2,4)

(-2,4]

(4,\infty)

(-\infty,4)

Explanation

4. Rational Inequalities

ExplanationZN · Zero of numerator

May be included when equality is allowed.

Common mistakeZD · Zero of denominator

Never included because the expression is undefined there.

Common mistakeImportant: do not cross-multiply by an expression containing

x

unless its sign is known.

Worked example

Guided 5

\frac{1+x}{1-x}\ge1

\frac{1+x}{1-x}-1\ge0

\frac{2x}{1-x}\ge0

ZN: $x=0$ . ZD: $x=1$ . Use the sign chart and remember that $x=1$ is never included.

Interval	x < 0	0 < x < 1	x > 1
Sign	−	+	−
Choose	No	Yes	No

[0,1)

Final Representation

[0, 1)

0

1

Worked example

Guided 6

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

0<\frac{x+1}{x-3}+2=\frac{3x-5}{x-3}

Critical numbers: $\frac53$ from the numerator and $3$ from the denominator.

Interval	x < 5/3	5/3 < x < 3	x > 3
Sign	+	−	+
Choose	Yes	No	Yes

(-\infty,\frac53)\cup(3,\infty)

Final Representation

(-∞, 5/3) ∪ (3, ∞)

←

→

5/3

3

Interactive lab

Rational sign chart laboratory

Build the sign chart for

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

. Select critical numbers, set signs, and choose the solution intervals.

1 · Select all critical numbers

Numerator zeros may be endpoints. Denominator zeros are always excluded.

2 · Cycle each sign: ? → + → −

3 · Select solution intervals

Quick Check

Check the idea before moving on.

Which critical numbers can NEVER be included?

\frac{4x^2}{x^2-9}\le0

Your turn

Which solution solves

\frac{x-1}{x+2}\ge0

?

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

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

[-2,1]

(-2,1]

Explanation

5. Absolute Value Inequalities

Absolute value is distance and can never be negative. Isolate the absolute-value expression first.

|u|<c\iff-c<u<c\quad\text{(AND · inside)}

|u|>c\iff u>c \text{or} u<-c\quad\text{(OR · outside)}

Worked example

Guided 7 · Isolate first

-|3x+2|+2<-2

-|3x+2|<-4

|3x+2|>4

3x+2>4\quad\text{or}\quad3x+2<-4

x>\frac23\quad\text{or}\quad x<-2

Final Representation

(-∞, -2) ∪ (2/3, ∞)

←

→

-2

2/3

Worked example

Guided 8 · Less than means AND

|2x-3|<4

-4<2x-3<4

-1<2x<7

-\frac12<x<\frac72

Final Representation

(-1/2, 7/2)

-1/2

7/2

Common mistakeNegative RHS:

|u|=-a\Rightarrow\varnothing

,

|u|>-a\Rightarrow(-\infty,\infty)

,

|u|<-a\Rightarrow\varnothing

for

a>0

.

Interactive lab

Absolute value inequality laboratory

Solve

|2x-3|<4

: choose inside or outside, then find both boundaries. Fractions and equivalent decimals are accepted.

1 · Inside or outside?

Less than a positive distance means being closer than that distance to the center.

2 · Boundaries in increasing order

Left endpointRight endpoint

Quick Check

Check the idea before moving on.

Without solving, which structure should be used?

|2x-3|<4

Your turn

Solve

|2x-3|\le5

.

[-1,4]

x\le-1\ \text{or}\ x\ge4

(-1,4)

[-4,1]

Practice

6. Mastery Checkpoint

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Linear inequalities & sign reversal

Compound inequalities

Nonlinear sign charts

Repeated factors

Rational inequalities · ZN vs ZD

Absolute-value inequalities

Rate all skills to prepare your personalized practice path.

Find your next lesson

Inequalities

1. Rules for Inequalities

Linear inequality builder

Your turn

Your turn · student questions

2. Linear & Compound Inequalities

Compound inequality builder

Interval and number line builder

3. Nonlinear Inequalities — Mr. Azzam’s Routine

Polynomial sign chart laboratory

Your turn

Your turn · student questions

4. Rational Inequalities

Rational sign chart laboratory

Your turn

Your turn · student questions

5. Absolute Value Inequalities

Absolute value inequality laboratory

Your turn

Your turn · student questions

6. Mastery Checkpoint

Keep your unfinished answers?