4. Equations & Inequalities

Lesson

We have looked at solving one step inequalities. We learned that the process is almost identical to that of solving equations, but we also need to keep in mind which operations cause the inequality symbol to reverse.

In particular, we found that **multiplying or dividing by a negative number** causes the inequality symbol to change direction. Also, **writing an inequality in reverse order** causes the inequality symbol to reverse.

Let's take a look at solving a slightly more complicated inequality, such as $-3x+2\ge14$−3`x`+2≥14. There are now two operations being applied to $x$`x` (multiplication and addition). Much like solving equations with two (or more) operations, we will need to take the order of operations into consideration as well.

Looking at the inequality $-3x+2\ge14$−3`x`+2≥14 and thinking about the order of operations, we can see that $x$`x` is first multiplied by $-3$−3 and then $2$2 is added. To solve this inequality, we want to undo these operations in **reverse order**. That is, we can solve this inequality by first subtracting $2$2 from both sides, then dividing both sides by $-3$−3 (which will change the inequality symbol used):

$-3x+2$−3x+2 |
$\ge$≥ | $14$14 |
Given |

$-3x+2-2$−3x+2−2 |
$\ge$≥ | $14-2$14−2 |
Subtracting $2$2 from both sides |

$-3x$−3x |
$\ge$≥ | $12$12 |
Simplifying |

$\frac{-3x}{-3}$−3x−3 |
$\le$≤ | $\frac{12}{-3}$12−3 |
Dividing both sides by $-3$−3 (and changing the inequality symbol) |

$x$x |
$\le$≤ | $-4$−4 |
Simplifying |

In this case, we arrive at the result $x\le-4$`x`≤−4. We can test some values in the original inequality to see if this is the right solution set - let's say $x=-5$`x`=−5 and $x=-3$`x`=−3.

- When $x=-5$
`x`=−5, we have $-3x+2=-3\times\left(-5\right)+2=17$−3`x`+2=−3×(−5)+2=17, which**is**greater than or equal to $14$14. - When $x=-3$
`x`=−3, we have $-3x+2=-3\times\left(-3\right)+2=11$−3`x`+2=−3×(−3)+2=11, which**is not**greater than or equal to $14$14.

So our result of $x\le-4$`x`≤−4 seems to be correct. We can graph this on the number line if required.

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.

Solve the following inequality: $3x+27>3$3`x`+27>3

Solve the following inequality: $\frac{a}{5}+3>3$`a`5+3>3

Consider the inequality $7-x>13$7−`x`>13.

Solve the inequality.

Now plot the solutions to the inequality $7-x>13$7−

`x`>13 on the number line below.

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

(b) Solve word problems leading to inequalities of the form px + q > r, px + q ≥ r, px + q ≤ r, or px + q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality on the number line and interpret it in the context of the problem.