courses › Algebra

Inequalities

level 23 course

Solve inequalities and count the whole numbers that fit.

Pen and paper is fine · no calculator needed why?

Opens at level 23.

Builds on: Linear equations (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

An inequality compares two sides: x > 3 means any number above 3. You solve it like an equation, with one extra step. Multiplying or dividing by a negative flips the sign, because it mirrors the numbers around zero: 2 is less than 5, but −2 is more than −5.

An absolute value such as |x − 3| is the distance from x to 3. So |x − 3| ≤ 5 means x is within 5 of 3.

Techniques

Solve, and flip for a negative

An inequality like ax + b > c.

Undo the number added or taken away, on both sides, as in an equation.
Divide both sides by the number in front of x.
If that number is negative, flip the sign: < becomes >, and ≤ becomes ≥.

worked example

Example: Solve −4x + 3 ≥ 23. What is the largest value x can take?

Take 3 from both sides: −4x ≥ 20.
Divide by −4 and flip the sign: x ≤ −5.
So x can be −5 or anything below it.

Answer: −5

Step to the nearest integer

The question asks for the largest or smallest integer that fits.

Solve for n. The boundary is usually not a whole number.
Largest n: step left along the number line to the first integer.
Smallest n: step right to the first integer.
Left means more negative: the first integer below −2.5 is −3.

worked example

Example: What is the largest integer n with 3n + 8 < 1?

Take 8 from both sides: 3n < −7.
Divide by 3: n < −7/3, about −2.33.
Step left to the first integer: −3.

Answer: −3

Count the integers within a distance

You count the integers x with |x − a| ≤ r or |x − a| < r.

The integers run from a − r to a + r.
With ≤, both ends count: last minus first, plus 1.
With <, both ends are left out, so there are 2 fewer.

worked example

Example: How many integers x satisfy |x + 2| < 4?

|x + 2| is the distance from x to −2.
Within 4 of −2, ends left out: −6 < x < 2.
The integers are −5 to 1: 1 − (−5) + 1 = 7.

Answer: 7

Tips by skill

TipSolve an inequality: Solve it like an equation. When you divide by a negative number, flip the inequality sign.
TipLargest whole solution: Solve for the boundary. For the largest integer, step left to the first integer; for the smallest, step right.
TipCount the solutions: The integers run from a − r to a + r. Count both ends with ≤; leave both ends out with <.

Watch out for

Forgetting to flip the sign after dividing by a negative number.
Giving the boundary, like 9.25, when the question asks for an integer.
For the largest integer, chopping the decimal off a negative boundary. Below −2.5 the answer is −3, not −2.
Counting the integers from −2 to 8 as 10. Both ends count, so there are 11.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Solve an inequality par 15 s not tried yet
worked example
Solve for x: 3x + 7 < −17
1. x < 8
2. x < −8
3. x > 8
4. x > −8
Answer: x < −8
1. Subtract 7 from both sides: 3x < −24.
2. Divide by 3; it is positive, so the sign stays: x < −8.
Largest whole solution par 16 s not tried yet
worked example
What is the largest integer n with 3n − 13 < −24?

Answer: -4
1. Add 13 to both sides: 3n < −11.
2. Divide by 3: n < −11/3, which is about −3.67.
3. The largest integer that fits is −4.
Count the solutions par 20 s not tried yet
worked example
How many integers x satisfy |x − 4| ≤ 2?

Answer: 5
1. |x − 4| ≤ 2 means −2 ≤ x − 4 ≤ 2, so 2 ≤ x ≤ 6.
2. Count both ends: 6 − 2 + 1 = 5 integers.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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