Rearrange so that one side is zero before finding the roots.
The roots divide the number line into intervals. Choose intervals where the quadratic has the required sign.
Use open endpoints for strict inequalities and closed endpoints when equality is included. A solution is a set of values, not just the roots.
Example 1
Solve the inequality using a sketch of the quadratic graph.
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is above the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.
Example 2
Solve the inequality using a sketch of the quadratic graph.
Subtract the right-hand term from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Include both roots because equality is allowed.
Sketch, not to scale. Roots and solution intervals are exact.
Your turn
Question 1
Solve the inequality using a sketch of the quadratic graph.
Check answer
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.
Question 2
Solve the inequality using a sketch of the quadratic graph.
Check answer
Subtract the right-hand term from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is above the -axis. Include both roots because equality is allowed.
Sketch, not to scale. Roots and solution intervals are exact.
Question 3
Solve the inequality using a sketch of the quadratic graph.
Check answer
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is negative, so the parabola opens downwards. Read where the curve is above the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.
Question 4
Solve the inequality using a sketch of the quadratic graph.
Check answer
Subtract the right-hand term from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Include both roots because equality is allowed.
Sketch, not to scale. Roots and solution intervals are exact.
Question 5
Solve the inequality using a sketch of the quadratic graph.
Check answer
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.
Solve the inequality using a sketch of the quadratic graph.
Answer
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.
Solve the inequality using a sketch of the quadratic graph.
Answer
Subtract the right-hand term from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is above the -axis. Include both roots because equality is allowed.
Sketch, not to scale. Roots and solution intervals are exact.
Solve the inequality using a sketch of the quadratic graph.
Answer
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is negative, so the parabola opens downwards. Read where the curve is above the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.
Solve the inequality using a sketch of the quadratic graph.
Answer
Subtract the right-hand term from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Include both roots because equality is allowed.
Sketch, not to scale. Roots and solution intervals are exact.
Solve the inequality using a sketch of the quadratic graph.
Answer
Subtract the right-hand constant from both sides, so the right-hand side becomes zero.Find the boundary values by solving the corresponding equation with zero.Factorise to find where the graph meets the -axis.Set each factor equal to zero: or . These values divide the number line into three intervals.The coefficient of is positive, so the parabola opens upwards. Read where the curve is below the -axis. Exclude both roots because the inequality is strict.
Sketch, not to scale. Roots and solution intervals are exact.