LEMMA

Equations & inequalities

Compound inequalities

ALEKS placement

“And” is an overlap; “or” is a union.

What this covers

  • A three-part inequality is two inequalities joined by “and”: −7 < 2x − 1 and 2x − 1 ≤ 5. Both must hold.
  • Work all three parts at once, or split into two lines and solve each. Splitting avoids careless slips.
  • Whatever you do to the middle, do to both ends. Dividing by a negative flips both symbols.
  • “And” keeps only the region both parts share. If the two pieces point away from each other, nothing survives.
  • “Or” keeps everything either part covers, even two pieces with a gap between them.
  • Keep each symbol as it was: one end can stay open while the other stays closed.

Worked example

Worked example

Solve -7 < 2x - 1 ≤ 5.

  1. −6<2x≤6Add 1 to all three parts.
  2. −3<x≤3Divide all three parts by 2.
Another worked example

Solve −15 < −3x + 6 < 21.

  1. −21<−3x<15Subtract 6 from all three parts.
  2. 7>x>−5Divide all three parts by −3. Both inequality signs reverse.
  3. −5<x<7Rewrite from least to greatest. It says the same thing.

A common mistake

A mistake Lemma catches

From −7<2x−1≤5

−7<2x≤6−6<2x≤6

“A three-part inequality changes in all three parts at once. Adding 1 reached the middle and the right (5 + 1 = 6) but not the left: −7 + 1 = −6, so −6 < 2x ≤ 6.”

Try one

Sample problem

Solve the compound inequality.

−2<x+4<9

Practice compound inequalities free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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