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Lesson · Grade 11 · Pre Calculus

Solving Linear Inequalities and Interval Notation

  • Grade 11
  • Pre Calculus
  • 2 min 9 s
  • British Columbia
  • BC11-MA-77

The video solves 3x − 7 ≤ −4 to get x ≤ 1, draws it as a closed dot shaded to the left, and writes it as (−∞, 1], explaining when to use square and round brackets. It then solves −2x + 5 > 11, shows the sign flipping when both sides are divided by −2 to give x < −3, proves it by testing x = −4 (true) and x = 0 (false), and writes the answer as (−∞, −3).

Concept: Solving linear inequalities and writing them in interval notation

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In one sentence

The question this lesson answers

An inequality compares two sides. Let us solve 3x minus 7 is less than or equal to negative 4.

Add 7 to both sides. Negative 4 plus 7 is 3, so 3x is less than or equal to 3.

Transcript

What the lesson says

An inequality compares two sides. Let us solve 3x minus 7 is less than or equal to negative 4.

Add 7 to both sides. Negative 4 plus 7 is 3, so 3x is less than or equal to 3.

Divide both sides by 3. 3 is positive, so the sign stays: x is less than or equal to 1.

A closed dot on the number line shows that 1 is included. Then shade everything to its left.

Interval notation writes a range by its two ends, left end first: negative infinity to 1.

A square bracket includes the end, and a round one does not. Infinity is not a number, so it always gets a round one.

Now solve negative 2x plus 5 is greater than 11.

Subtract 5 from both sides. 11 minus 5 is 6, so negative 2x is greater than 6.

Divide both sides by negative 2. Dividing by a negative flips the sign, so x is less than negative 3.

Test x equals negative 4. Negative 2 times negative 4 is 8, plus 5 is 13. 13 is greater than 11: true.

Test x equals 0. Negative 2 times 0 is 0, plus 5 is 5. 5 is not greater than 11: false.

Forgetting to flip is a common mistake: x greater than negative 3 would include 0, which just failed.

Now an open dot shows that negative 3 is not included. Shade to the left again.

In interval notation that is negative infinity to negative 3, with round brackets at both ends.

Recap: flip the sign when you multiply or divide by a negative, and use a square bracket only for an included end.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Inequalities: linear and quadratic inequalitiesBC11-MA-77

Next: Solving a Quadratic Inequality with Sign Analysis

The outcomes and strands follow the British Columbia curriculum.

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