An **inequality** shows the relationship between two unequal values.

Consider the inequality: 9 \lt 15

Add 3 to both sides of the inequality.

Add -3 from both sides of the inequality.

Subtract 3 to both sides of the inequality.

Subtract -3 from both sides of the inequality.

Multiply both sides of the inequality by 3.

Multiply both sides of the inequality by -3.

Divide both sides of the inequality by 3.

Divide both sides of the inequality by -3.

Was the inequaltiy true after each operation? If not, what could you do to make it true?

Test this out with other numbers. Is it still true?

Solving inequalities is very similar to solving equations except when multiplying or dividing by a negative.

Instead of using the properties of equality like we do for equations, when solving inequalities we use the ** properties of inequality **:

\text{Addition property of inequality} | \text{If } a\lt b, \text{then } a+c\lt b+c \\ \text{If } a \gt b, \text{then } a+c \gt b+c |
---|---|

\text{Subtraction property of inequality} | \text{If } a \lt b, \text{then } a-c \lt b-c \\ \text{If } a \gt b, \text{then } a-c \gt b-c |

\text{Multiplication property of inequality} | \text{If } a \lt b \text{ and } c \gt 0, \text{then } a \cdot c \lt b \cdot c \\ \text{If } a \gt b \text{ and } c \gt 0, \text{then } a \cdot c \gt b \cdot c |

\text{Multiplication property of inequality} \\\text{ (by a negative number)} | \text{If } a \lt b \text{ and } c \lt 0, \text{then } a \cdot c \gt b \cdot c \\ \text{If } a \gt b \text{ and } c \lt 0, \text{then } a\cdot c \lt b \cdot c |

\text{Division property of inequality} | \text{If } a \lt b \text{ and } c \gt 0, \text{then } \dfrac{a}{c} \lt \dfrac{b}{c} \\ \text{If } a \gt b \text{ and } c\gt 0, \text{then } \dfrac{a}{c} \gt \dfrac{b}{c} |

\text{Division property of inequality} \\\text{ (by a negative number)} | \text{if } a \lt b \text{ and } c\lt 0, \text{then } \dfrac{a}{c}\gt \dfrac{b}{c} \\ \text{if } a\gt b \text{ and } c\lt 0, \text{then } \dfrac{a}{c}\lt \dfrac{b}{c} |

Let's compare the equation -2x=8 and the inequality -2x\gt 8.

We will start by solving the equation -2x=8

\displaystyle \frac{-2x}{-2} | \displaystyle = | \displaystyle \frac{8}{-2} | Division property of equality (divide by -2) |

\displaystyle x | \displaystyle = | \displaystyle -4 | Evaluate the division |

This equation has exactly one solution, x=-4. We can graph the solution of -2x=8 on the number line like this:

This number line shows that there is only one solution to the equation, x=-4.

Next, let's solve the inequality -2x \gt 8

\displaystyle \frac{-2x}{-2} | \displaystyle < | \displaystyle \frac{8}{-2} | Division property of inequality (reverse the inequality symbol) |

\displaystyle x | \displaystyle < | \displaystyle -4 | Evaluate the division |

We can graph the solution of -2x \gt 8 on the number line like this:

This number line shows that there are many solutions to the inequality, including any number that is less than, but not equal to, -4.

We can verify solutions algebraically, by substituting the value of x back into the equation or inequality.

Let's verify the solution to the equation -2x=8. We found that the solution was x=-2.

\displaystyle -2\left(-4\right) | \displaystyle = | \displaystyle 8 | Substitution |

\displaystyle 8 | \displaystyle = | \displaystyle 8 | Evaluate the multiplication |

After substituting the value and evaluating the equation, we are left with 8=8. This is true, so the answer is correct.

To determine if a value is a solution to the inequality, we can substitute values to determine if they are true. Inequalities can have multiple solutions, so let's see if x=-2 satisfies the inequality -2x \gt 8.

\displaystyle -2\left(-2\right) | \displaystyle > | \displaystyle 8 | Substitution property |

\displaystyle 4 | \displaystyle > | \displaystyle 8 | Evaluate the multiplication |

We can read this as "4 is greater than 8" which is not true. x=-2 is not a solution to the inequality -2x\gt 8.

Consider the inequality 1+x\lt 2.

a

Solve the inequality for x.

Worked Solution

b

Now plot the solutions to the inequality 1+x \lt 2 on the number line below.

Worked Solution

Consider the inequality -4 \lt 2m.

a

Solve the inequality.

Worked Solution

b

Now plot the solutions to the inequality -4 \lt 2m on the number line below.

Worked Solution

Determine whether c=-2 satisfies the inequality c+3 \lt 4.

Worked Solution

Solve -\dfrac{x}{2} \geq 5 for x.

Worked Solution

Compare the process for solving 4.5x=27 and 4.5x\gt 27 and their solution sets.

Worked Solution

You have \$18 and you want to buy some slices of cake.

a

If a slice of cake costs \$5, write an inequality to find how many cake slices you can buy.

Worked Solution

b

Solve the inequality you wrote in part (a).

Worked Solution

Idea summary

The properties of inequality are used when solving inequalities:

\text{Addition property of inequality} | \text{If } a\lt b, \text{then } a+c\lt b+c \\ \text{If } a \gt b, \text{then } a+c \gt b+c |
---|---|

\text{Subtraction property of inequality} | \text{If } a \lt b, \text{then } a-c \lt b-c \\ \text{If } a \gt b, \text{then } a-c \gt b-c |

\text{Multiplication property of inequality} | \text{If } a \lt b \text{ and } c \gt 0, \text{then } a \cdot c \lt b \cdot c \\ \text{If } a \gt b \text{ and } c \gt 0, \text{then } a \cdot c \gt b \cdot c |

\text{Multiplication property of inequality} \\\text{ (by a negative number)} | \text{If } a \lt b \text{ and } c \lt 0, \text{then } a \cdot c \gt b \cdot c \\ \text{If } a \gt b \text{ and } c \lt 0, \text{then } a\cdot c \lt b \cdot c |

\text{Division property of inequality} | \text{If } a \lt b \text{ and } c \gt 0, \text{then } \dfrac{a}{c} \lt \dfrac{b}{c} \\ \text{If } a \gt b \text{ and } c\gt 0, \text{then } \dfrac{a}{c} \gt \dfrac{b}{c} |

\text{Division property of inequality} \\\text{ (by a negative number)} | \text{if } a \lt b \text{ and } c\lt 0, \text{then } \dfrac{a}{c}\gt \dfrac{b}{c} \\ \text{if } a\gt b \text{ and } c\lt 0, \text{then } \dfrac{a}{c}\lt \dfrac{b}{c} |

Remember, when multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality symbol so the inequality stays true.