Formulas and Tables

Elementary Algebra

Inequalities

Real numbers: \(a\), \(b\), \(c\), \(d\), \(x\), \(m\), \(n\)

Real positive numbers: \({a_1}\), \({a_2}\), …, \({a_n}\)

  1. Inequalities and intervals of the number line
Inequalities and intervals of the number line
  1. Strict inequalities
    The notation \(a \lt b\) means that “\(a\) is less than \(b\)”,
    The notation \(a \gt b\) means that “\(a\) is greater than \(b\)”.
  2. Non-strict inequalities
    The notation \(a \le b\) means that “\(a\) is less than or equal to \(b\)”,
    The notation \(a \ge b\) means that “\(a\) is greater than or equal to \(b\)”.
  3. If \(a \gt b\), then \(b \lt a\).
  4. If \(a \gt b\), then \(a – b \gt 0\) or equivalently \(b – a \lt 0\).
  5. Transitivity property
    If\(a \gt b\) and \(b \gt c\), then \(a \gt c\).
  6. Adding the same number to the both sides of an inequality does not change the sign of the inequality:
    If \(a \gt b\), then \(a + c \gt b + c\).
  7. Moving any term of an inequality from one side to another and changing its sign results in an equivalent inequality:
    If \(a + b \gt c\), then \(a \gt c – b\).
  8. If \(a \gt b\) and \(c \gt d\), then \(a + c \gt b + d\).
  9. If \(a \gt b\) and \(c \gt d\), then \(a – d \gt b – c\).
  10. Multiplying both sides of an inequality by the same positive number does not change the sign of the inequality:
    If \(a \gt b\) and \(m \gt 0\), then \(ma \gt mb\).
  11. Dividing both sides of an inequality by the same positive number does not change the sign of the inequality:
    If \(a \gt b\) and \(m \gt 0\), then \(a/m > b/m\).
  12. Multiplying both sides of an inequality by the same negative number inverts the inequality:
    If \(a \gt b\) and \(m \lt 0\), then \(ma \lt mb\).
  13. Dividing both sides of an inequality by the same negative number inverts the inequality:
    If \(a \gt b\) and \(m \lt 0\), then \(a/m \lt b/m\).
  14. If \(a \gt b \gt 0\), then \(1/b \gt 1/a\).
  15. Multiplication of inequalities
    If \(a \gt b \gt 0\) and \(c \gt d \gt 0\), then \(ac \gt bd\).
  16. Division of inequalities
    If \(a \ge b \gt 0\) and \(c \gt d \gt 0\), then \(a/d \gt b/c\).
  17. Raising an inequality to a positive power
    If \(a \gt b \gt 0\) and \(n \gt 0\), then \({a^n} \gt {b^n}\).
  18. Raising an inequality to a negative power
    If \(a \gt b \gt 0\) and \(n \lt 0\), then \({a^n} \lt {b^n}\).
  19. \(N\)th root of an inequality
    If \(a \gt b \gt 0\) and \(n \gt 0\), then \(\sqrt[\large n\normalsize]{a} \gt \sqrt[\large n\normalsize]{b}\).
  20. \(a + \large\frac{1}{a}\normalsize \ge 2\;\) \(\left( {a \gt 0} \right)\)
    The equality holds only if \(a = 1\).
  21. Cauchy’s inequality of arithmetic and geometric means
    \(\sqrt {ab} \le \left( {a + b} \right)/2,\) where \(a \gt 0,\) \(b \gt 0\).
    The equality holds only when \(a = b\).
  22. Cauchy’s inequality of arithmetic and geometric means (the case of several variables)
    \(\sqrt[n]{{{a_1}{a_2} \cdots {a_n}}} \le\) \( {\large\frac{{{a_1} + {a_2} + \ldots + {a_n}}}{n}\normalsize},\) where \({a_1},{a_2}, \ldots ,{a_n} \gt 0\).
  23. Linear inequality (case \(a \gt 0\))
    If \(ax + b \gt 0\) and \(a \gt 0\), then \(x \gt -b/a\).
  24. Linear inequality (case \(a \lt 0\))
    If \(ax + b \gt 0\) and \(a \lt 0\), then \(x \lt -b/a\).
  25. Quadratic inequality \(a{x^2} + bx + c \gt 0\)
Graphical solutions of quadratic inequalities
  1. \(\left| {a + b} \right| \le \left| a \right| + \left| b \right|\)
  2. If \(\left| x \right| \lt a\), then \(-a \lt x \lt a,\) where \(a \gt 0\).
  3. If \(\left| x \right| \gt a\), then \(x \lt -a\) and \(x \gt a,\) where \(a \gt 0\).
  4. If \({x^2} \lt a\), then \(\left| x \right| \lt \sqrt a \), where \(a \gt 0\).
  5. If \({x^2} \gt a\), then \(\left| x \right| \gt \sqrt a \), where \(a \gt 0\).
  6. \({\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \gt 0,\;\) \(\Leftrightarrow \;f\left( x \right)g\left( x \right) \gt 0\;\) \(\Leftrightarrow
    \;\begin{cases}
    {f\left( x \right)} \gt 0 \\
    {g\left( x \right)} \gt 0
    \end{cases}\;\) \(\text { or }\;
    \begin{cases}
    {f\left( x \right)} \lt 0 \\
    {g\left( x \right)} \lt 0
    \end{cases}.
    \)
  7. \({\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \lt 0,\;\) \(\Leftrightarrow \;f\left( x \right)g\left( x \right) \lt 0\;\) \(\Leftrightarrow
    \;\begin{cases}
    {f\left( x \right)} \gt 0 \\
    {g\left( x \right)} \lt 0
    \end{cases}\;\) \(\text { or }\;
    \begin{cases}
    {f\left( x \right)} \lt 0 \\
    {g\left( x \right)} \gt 0
    \end{cases}.
    \)
  8. \({\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \ge 0,\;\) \(\Leftrightarrow \;
    \begin{cases}
    {f\left( x \right) g\left( x \right)} \ge 0 \\
    {g\left( x \right)} \ne 0
    \end{cases}\;\) \(\Leftrightarrow
    \;\begin{cases}
    {f\left( x \right)} \ge 0 \\
    {g\left( x \right)} \gt 0
    \end{cases}\;\) \(\text { or }\;
    \begin{cases}
    {f\left( x \right)} \le 0 \\
    {g\left( x \right)} \lt 0
    \end{cases}.
    \)
  9. \({\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} \le 0,\;\) \(\Leftrightarrow \;
    \begin{cases}
    {f\left( x \right) g\left( x \right)} \le 0 \\
    {g\left( x \right)} \ne 0
    \end{cases}\;\) \(\Leftrightarrow
    \;\begin{cases}
    {f\left( x \right)} \le 0 \\
    {g\left( x \right)} \gt 0
    \end{cases}\;\) \(\text { or }\;
    \begin{cases}
    {f\left( x \right)} \ge 0 \\
    {g\left( x \right)} \lt 0
    \end{cases}.
    \)