PrerequisitesPrerequisite

Solving compound inequalities - with unknowns in all three parts

Solve compound inequalities with unknowns in all three parts.

  • Split the chain into two adjacent inequalities.
  • Solve each inequality; reverse its sign when dividing by a negative number.
  • Take the overlap and write the smaller bound first.

Example 1

Solve the compound inequality.

First inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Second inequality

Both inequalities must hold: take the overlap.

Values between -3 and 4. -3 included, 4 excluded.

Example 2

Solve the compound inequality.

First inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Second inequality

Both inequalities must hold: take the overlap.

Values between 3 and 4. 3 excluded, 4 included.

Remember
The final interval must satisfy both inequalities.

Your turn

Question 1

Solve the compound inequality.

Check answer

First inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Second inequality

Both inequalities must hold: take the overlap.

Values between -2 and 5. -2 included, 5 excluded.
Question 2

Solve the compound inequality.

Check answer

First inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Second inequality

Both inequalities must hold: take the overlap.

Values between 1 and 6. 1 excluded, 6 included.
Question 3

Solve the compound inequality.

Check answer

First inequality

Second inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Both inequalities must hold: take the overlap.

Values between -4 and 2. -4 included, 2 included.
Question 4

Solve the compound inequality.

Check answer

First inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Second inequality

Both inequalities must hold: take the overlap.

Values between -5 and -1. -5 excluded, -1 excluded.
Question 5

Solve the compound inequality.

Check answer

First inequality

Reverse the inequality sign because both sides are multiplied or divided by a negative number.

Second inequality

Both inequalities must hold: take the overlap.

Values between 0 and 3. 0 included, 3 excluded.
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