Inequalities
Solve inequalities and count the whole numbers that fit.
Pen and paper is fine · no calculator needed why?
Opens at level 23. You're level 1. You can read and practice here now.
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
- 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
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
- 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
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
- 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
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
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.
practice
Solve an inequality
worked example
Solve for x: 3x + 7 < −17
- x < 8
- x < −8
- x > 8
- x > −8
Answer: x < −8
- Subtract 7 from both sides: 3x < −24.
- Divide by 3; it is positive, so the sign stays: x < −8.
Largest whole solution
worked example
What is the largest integer n with 3n − 13 < −24?
Answer: -4
- Add 13 to both sides: 3n < −11.
- Divide by 3: n < −11/3, which is about −3.67.
- The largest integer that fits is −4.
Count the solutions
worked example
How many integers x satisfy |x − 4| ≤ 2?
Answer: 5
- |x − 4| ≤ 2 means −2 ≤ x − 4 ≤ 2, so 2 ≤ x ≤ 6.
- Count both ends: 6 − 2 + 1 = 5 integers.