Solving inequalities

• What is a linear inequality?

  An inequality that tells you that one expression is greater than or less than another.
• What does “⩾” mean?

  It means “greater than or equal to”.
• What does “⩽” mean?

  It means “less than or equal to”.
• What are the components of a linear inequality?


  A linear inequality has constant terms and terms in x (and/or y), but no higher powers of x.
• Why is 3x > 12 not a linear inequality?

  Because it contains a term with x raised to a power greater than one.
• How would you read the inequality 3x + 4 ⩾ 7?


  It would be read as “3x + 4 is greater than or equal to 7”.
• How do you solve linear inequalities?

  By following the same rules as solving linear equations while keeping the inequality sign.
• What happens when you multiply or divide both sides of an inequality by a negative number?

  You must flip the sign of the inequality.
• Why should you never multiply or divide by a variable (x) when solving inequalities?

  Because the variable could be positive or negative.

  
• What is the safest way to rearrange a linear inequality?

  By adding and subtracting to move all terms onto one side.
• How do you represent linear inequalities on a number line?


  • Use an open circle and an arrow for x < a (arrow points left) and x > a (arrow points right).
  • Use a solid circle and an arrow for x ≤ a (arrow points left) and x ≥ a (arrow points right).
  • Use two circles for a < x < b (open circles) and a ≤ x ≤ b (solid circles) with a line between them.
  • For disjoint inequalities like "x < a or x > b", use two circles at a and b with arrows pointing away from each.
• What does the notation {x: ...} mean in set notation?

  It means "x is in the set ...".
• How would you express x > 3 in set notation?


  {x: x > 3}

  
• How do you express x between two values in set notation?

  By writing the two end values in separate sets using the intersection symbol, ∩.

  
• How would you express x > 3 and x ≤ 5 in set notation?

  {x: x > 3} ∩ {x: x ≤ 5}

  
• How do you express disjoint inequalities in set notation?

  By writing the two end values in separate sets using the union symbol, ∪.

  
• How would you express x < 3 or x ≥ 5 in set notation?

  {x: x < 3} ∪ {x: x ≥ 5}

  
• How do you solve double inequalities?

  By doing the same operation to all three parts of the inequality.
• What should you not do when solving linear inequalities?

  Do not change the inequality sign to an equals sign.
• What is the first step in solving the double inequality −7 ≤ 3x − 1 < 2?

  Add 1 to all three parts.

  
• What should you remember when dividing all parts of an inequality by a positive number?

  There is no need to flip the signs.
• How do you illustrate the final answer of a double inequality on a number line?

  Using an open circle at 1 and a closed circle at -2.
• How do you solve the inequality 5 - 2x ≤ 21?

  Subtract 5 from both sides, then divide by -2, flipping the inequality sign.

  
• What is the first step in solving quadratic inequalities?

  Rearrange the inequality into quadratic form with a positive squared term.
• What shape will the graph of a quadratic inequality have if the squared term is positive?

  It will be "U" shaped.
• What do you need to identify after sketching the graph of a quadratic inequality?

  The region that satisfies the inequality.
• For ax + bx + c > 0, where do you want the solution to be?

  The region above the x-axis.
• For ax + bx + c < 0, where do you want the solution to be?

  The region below the x-axis.
• What should you avoid when solving quadratic inequalities?

  Avoid multiplying or dividing by a negative number unless you flip the inequality sign.
• What is the exam tip for solving quadratic inequalities?

  Always start by rearranging to a quadratic with a positive squared term and sketch a graph before deciding the final answer.