Let's look at the inequality -3x+2 \geq 14 and think about the order of operations.

We can see that the operations applied to the variable x are:

First multiply by -3

Next add 2

To solve this inequality, we want to undo these operations in reverse order. So we will:

First, subtract 2 from both sides

Next, divide both sides by -3 (and reverse the inequality symbol)

All together this looks like:

\displaystyle -3x+2 | \displaystyle \geq | \displaystyle 14 | Given inequality |

\displaystyle -3x+2-2 | \displaystyle \geq | \displaystyle 14-2 | Subtraction 2 from both sides |

\displaystyle -3x | \displaystyle \geq | \displaystyle 12 | Evaluate the subtraction |

\displaystyle \dfrac{-3x}{-3} | \displaystyle \leq | \displaystyle \dfrac{12}{-3} | Divide both sides by -3 (reverse the inequality sign) |

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

We found that x\leq-4. We can test some values in the given inequality to see if they result in a true statement. Let's try the numbers just above and below -4, say x=-5 and x=-3.

When x=-5, we have -3x+2 = -3 \cdot \left(-5\right) +2 = 17, which is greater than or equal to 14.

When x=-3, we have -3x+2 = -3 \cdot \left(-3\right) +2 = 11, which is not greater than or equal to 14.

So our result of x\leq-4 seems to be correct. We can also graph this on the number line. For x\leq-4, we will include -4 on the number line as a filled circle and a ray pointing to the left side.

Let's compare how we solved that inequality to how we solve a similar equation, -3x+2=14.

\displaystyle -3x+2 | \displaystyle = | \displaystyle 14 | Given equation |

\displaystyle -3x+2-2 | \displaystyle = | \displaystyle 14-2 | Subtraction 2 from both sides |

\displaystyle -3x | \displaystyle = | \displaystyle 12 | Evaluate the subtraction |

\displaystyle \dfrac{-3x}{-3} | \displaystyle = | \displaystyle \dfrac{12}{-3} | Divide both sides by -3 |

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

We can check the solution to the equation using the substitution property.

\displaystyle -3\left(-4\right)+2 | \displaystyle = | \displaystyle 14 | Substitution property |

\displaystyle 12+2 | \displaystyle = | \displaystyle 14 | Evaluate the multiplication |

\displaystyle 14 | \displaystyle = | \displaystyle 14 | Evaluate the addition |

Because 14=14, our solution is correct. We can graph the solution to the equation as a single point on a number line like this:

This graph differs from the graph of the solution to -3x+2\geq 14 because the solution to the equation 3x+2=14 is only one value while the solution set for the inequality contains an infinite number of values.

Given 3x+27\gt 3

a

Solve the inequality.

Worked Solution

b

Determine if -8 is a solution to the inequality.

Worked Solution

Solve the following inequality: 4.2 \lt \dfrac{a}{5.2} + 3.1

Worked Solution

Consider the inequality -7-x \gt 13.

a

Solve the inequality.

Worked Solution

b

Now, plot the solutions to the inequality -7-x \gt 13 on a number line.

Worked Solution

In an ecological park, you were assigned to manage the number of people who can go through the underground cave. According to safety guidelines, no more than 10 people can be in the cave at a time. Write an inequality to represent the number of people that can go through the underground cave at a time, given that there are 2 accompanying guides for each group going.

Worked Solution

Idea summary

When solving any inequality:

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

Writing an inequality in reverse order also reverses the inequality symbol.

When solving an inequality with two (or more) operations:

It is generally easiest to undo one operation at a time, in reverse order to the order of operations.