3. Equations & Inequalities

Lesson

In most cases when we have an inequality involving a variable, it will be useful to simplify and solve the inequality.

We are used to solving equations, and the method for solving inequalities is very similar. Most importantly, "whatever operation is done to one side needs to be done to the other side" is still true for inequalities!

In fact, the steps that we take are the **same steps** as for equations, but with one extra consideration: **which way does the inequality symbol face after each step**?

Addition:

Numeric | Algebraic | |||||

$3$3 | $<$< | $4$4 | $x-2$x−2 |
$<$< | $4$4 | |

$3+2$3+2 | $<$< | $4+2$4+2 | $x-2+2$x−2+2 |
$<$< | $4+2$4+2 | |

$5$5 | $<$< | $6$6 | $x$x |
$<$< | $6$6 |

As we can see, adding a number to both sides of an inequality **doesn't change** the inequality symbol.

Subtraction:

Numeric | Algebraic | |||||

$5$5 | $>$> | $2$2 | $x+1$x+1 |
$\ge$≥ | $2$2 | |

$5-1$5−1 | $>$> | $2-1$2−1 | $x+1-1$x+1−1 |
$\ge$≥ | $2-1$2−1 | |

$4$4 | $>$> | $1$1 | $x$x |
$\ge$≥ | $1$1 |

Similarly, subtracting a number from both sides of an inequality **doesn't change** the inequality symbol.

Multiplication by positive numbers:

Numeric | Algebraic | |||||

$4$4 | $<$< | $7$7 | $\frac{x}{3}$x3 |
$\le$≤ | $7$7 | |

$4\times3$4×3 | $<$< | $7\times3$7×3 | $\frac{x}{3}\times3$x3×3 |
$\le$≤ | $7\times3$7×3 | |

$12$12 | $<$< | $21$21 | $x$x |
$\le$≤ | $21$21 |

Multiplication by negative numbers:

Numeric | Algebraic | |||||

$4$4 | $<$< | $7$7 | $-\frac{x}{3}$−x3 |
$<$< | $7$7 | |

$4\times\left(-3\right)$4×(−3) | $>$> | $7\times\left(-3\right)$7×(−3) | $\left(-\frac{x}{3}\right)\times\left(-3\right)$(−x3)×(−3) |
$>$> | $7\times\left(-3\right)$7×(−3) | |

$-12$−12 | $>$> | $-21$−21 | $x$x |
$>$> | $-21$−21 |

Now we have found a difference! Multiplying both sides of an inequality by a **positive** number **doesn't change** the inequality symbol, but multiplying by a **negative** number **does** change the inequality symbol.

Division:

Remember that we can think about division as an equivalent multiplication. For example, dividing by $3$3 is the same as multiplying by $\frac{1}{3}$13. So we get the same result: dividing both sides of an inequality by a **positive** number **doesn't change** the inequality symbol, but dividing by a **negative** number **does** change the inequality symbol.

It is often useful to write an equation in reverse order. For example, if we reach the step $3=x$3=`x` in our work, we often reverse this and write $x=3$`x`=3.

We can also do this with inequalities, as long as we **reverse the symbol** as well. For example, "$3$3 is greater than $x$`x`" means the same thing as "$x$`x` is less than $3$3". So $3>x$3>`x` can be rewritten as $x<3$`x`<3, using the opposite symbol.

Same symbol

The following operations don't change the inequality symbol used:

**Adding**a number to both sides of an inequality.**Subtracting**a number from both sides of an inequality.**Multiplying**both sides of an inequality by a**positive**number.**Dividing**both sides of an inequality by a**positive**number.

Opposite symbol

The following operations reverse the inequality symbol used:

- Writing an inequality in reverse order.
**Multiplying**both sides of an inequality by a**negative**number.**Dividing**both sides of an inequality by a**negative**number.

Solve the following inequality: $x+5>14$`x`+5>14

Solve the following inequality: $x+5\ge11$`x`+5≥11

Solve the following inequality: $10x<90$10`x`<90

Recall that we can plot inequalities by using number lines.

For example, a plot of the inequality $x\le4$`x`≤4 looks like this:

Now let's consider an inequality such as $x+3>5$`x`+3>5. What would we plot for this inequality?

As in the case of $x\le4$`x`≤4 above, what we want to plot on the number line are all of the possible values that the variable can take - that is, the **solutions** of the inequality. The inequality $x+3>5$`x`+3>5 has the solutions "all numbers which, when added to $3$3 result in a number greater than $5$5". This is a little bit of a mouthful already, and there are definitely much more complicated inequalities than this!

So in order to plot the solutions to an inequality such as $x+3>5$`x`+3>5, it will be easiest to first solve the inequality. In this case, we can subtract $3$3 from both sides to get $x>2$`x`>2. So the plot will show "all numbers greater than $2$2" on the number line, which looks like this:

Remember

When solving an inequality:

- Multiplying or dividing both sides by a
**negative**number will reverse the inequality symbol. - Reversing the order of the inequality will reverse the inequality symbol too.

When plotting an inequality:

- The symbols $<$< and $>$>
**don't**include the end point, which we show with a**hollow**circle. - The symbols $\ge$≥ and $\le$≤
**do**include the endpoint, which we show with a**filled**circle.

Consider the inequality $3+x<2$3+`x`<2.

Solve the inequality for $x$

`x`.Now plot the solutions to the inequality $3+x<2$3+

`x`<2 on the number line below. Make sure to use the correct type of endpoint.

Consider the inequality $2x>-4$2`x`>−4.

Solve the inequality.

Now plot the solutions to the inequality $2x>-4$2

`x`>−4 on the number line below.

Consider the inequality $\frac{x}{-7}<2$`x`−7<2.

Solve the inequality.

Now plot the solutions to the inequality $\frac{x}{-7}<2$

`x`−7<2 on the number line below.