The inequality operator compares the quantities of two values and returns a boolean.

Variants

• ${\displaystyle a > b}$ Returns true if a is greater than b, else return false
• ${\displaystyle a < b}$ Returns true if a is less than b, else return false
• ${\displaystyle a \geq b}$ Returns true if a is greater than b or equal to b, else return false
• ${\displaystyle a \le b}$ Returns true if a is less than b or equal to b, else return false

For example ${\displaystyle 6>5}$is true because 6 is farther right on the number line than 5.

Properties

All previously mentioned variants of the inequality operator share most of the same properties. (For ${\displaystyle x<0}$on the multiplicative property negate the result for ${\displaystyle x>0}$)

• Additive property - If ${\displaystyle a > b}$, then ${\displaystyle a + x > b + x}$
• Multiplicative property - If ${\displaystyle a > b}$, and if ${\displaystyle x>0}$then ${\displaystyle ax> bx}$else if ${\displaystyle x<0}$then ${\displaystyle ax < bx}$
• Exponential Property - If ${\displaystyle a > b}$, and if ${\displaystyle x>1}$then ${\displaystyle x^a > x^b }$else if ${\displaystyle 0<x<1}$then ${\displaystyle x^a < x^b }$

Other

• ${\displaystyle a \geq b = a > b }$or ${\displaystyle a = b}$
• Not ${\displaystyle a > b = a \leq b }$
• If ${\displaystyle a > b}$, then ${\displaystyle b<a}$
• If ${\displaystyle a < b}$, and ${\displaystyle b < c}$, then ${\displaystyle a < c }$
• If ${\displaystyle a < b}$, and ${\displaystyle a < c }$, then ${\displaystyle a < \min(b,c) }$

Calculation

Basics

${\displaystyle \begin{align} 3x - 5 & > 4 \\ 3x - 5 + 5 & > 4 + 5 \\ 3x & > 9 \\ \frac {3x}3 & > \frac 93 \\ x & > 3 \end{align} }$

Negative Factors

${\displaystyle \begin{align} -\frac 23x & > 4 \\ -2x & > 12 \\ -x & > 6 \\ x & < 6 \end{align} }$

Exponential Inequalities

${\displaystyle \begin{align} \sqrt{3} \cdot 3^x & < \frac {18^x}{2^x} \\ 3^{\frac 12} \cdot 3^x & < \frac {2^x \cdot 9^x}{2^x} \\ 3^{(x+ \frac 12)} & < 9^x \\ 3^{(x+ \frac 12)} & < 3^{2x} \\ x + \frac 12 & < 2x \\ \frac 12 & < x \end{align}}$

Trinomial Inequalities

${\displaystyle \begin{align} -5x^2 + 15x & > -20 \\ -5x^2 + 15x + 20 & > 0 \\ x^2 - 3x - 4 & > 0 \\ x^2 - 4x + x - 4 & > 0 \\ x(x - 4) + 1(x - 4) & > 0 \\ (x + 1)(x - 4) & > 0 \end{align} }$

${\displaystyle \begin{cases} \text{if} \quad x - 4 > 0, & \text{then} \quad x + 1 > 0 \\ \text{if} \quad x - 4 < 0, & \text{then} \quad x + 1 < 0 \end{cases} }$

${\displaystyle \begin{cases} \text{if} \quad x > 4, & \text{then} \quad x > -1 \\ \text{if} \quad x < 4, & \text{then} \quad x < -1 \end{cases} }$

${\displaystyle \begin{align} x > 4 & \quad \text{and} \quad x > -1 \\ \text{or} \quad x < 4 & \quad \text{and} \quad x < -1 \end{align} }$

${\displaystyle \begin{align} && x > \max(4, -1) \\ \text{or} && x < \min(4, -1) \end{align} }$

${\displaystyle x > 4 \quad \text{or} \quad x < -1 }$

Faster:

${\displaystyle \begin{align} x^2 + 2x - 8 & < 0 \\ (x+4)(x-2) & < 0 \\ \\ x + 4 > 0 \quad \text{and} \quad x - 2 & < 0 \\ \text{or} \quad x + 4 < 0 \quad \text{and} \quad x - 2 & > 0 \\ x > -4 \quad \text{and} \quad x & < 2 \\ \text{or} \quad x < -4 \quad \text{and} \quad x & > 2 \\ -4 < x & < 2 \end{align} }$

Graphing Inequalities

When graphing inequalities, be sure that if the equation is x > 4, the circle is open and the arrow points to the right.

If the equation is x < 4, however, the circle is open and the arrow points to the left.

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