4. Equations & Inequalities

Lesson

When we manipulate with equations, we can apply the same operation to both sides and the equation remains true. Consider the following equation:

$x+7$x+7 |
$=$= | $12$12 |

We can subtract $7$7 from both sides of the equation in order to find the value of $x$`x`. This is because both sides of the equation are identical, so what we do to one side, we should do to the other side.

$x+7$x+7 |
$=$= | $12$12 |
(rewriting the equation) |

$x+7-7$x+7−7 |
$=$= | $12-7$12−7 |
(subtracting $7$7 from both sides) |

$x$x |
$=$= | $5$5 |
(simplifying both sides) |

When working with inequalities, this is not necessarily always the case.

Consider the inequality $9<15$9<15.

If we add or subtract both sides by any number, say $3$3, we can see that the resulting inequality remains true.

$9+3<15+3$9+3<15+3 $12<18$12<18 |

$9-3<15-3$9−3<15−3 $6<12$6<12 |

Note that adding a negative would have the same effect as subtracting, so we can also add and subtract negative numbers without changing the inequality.

What happens if we multiply or divide both sides?

Multiplying by a positive

$9$9 | $<$< | $15$15 |
Given |

$9\times3$9×3 | $<$< | $15\times3$15×3 |
Multiplying by $3$3 |

$27$27 | $<$< | $45$45 |
Simplifying |

The inequality stays the same

Dividing by a positive

$9$9 | $<$< | $15$15 |
Given |

$\frac{9}{3}$93 | $<$< | $\frac{15}{3}$153 |
Dividing by $3$3 |

$3$3 | $<$< | $5$5 |
Simplifying |

The inequality stays the same

Multiplying by a negative

$9$9 | $<$< | $15$15 |
Given |

$9\times\left(-3\right)$9×(−3) | $>$> | $15\times\left(-3\right)$15×(−3) |
Multiplying by $-3$−3 |

$-27$−27 | $>$> | $-45$−45 |
Simplifying |

The inequality switches from $<$< to $>$>!

Dividing by a negative

$9$9 | $<$< | $15$15 |
Given |

$\frac{9}{-3}$9−3 | $>$> | $\frac{15}{-3}$15−3 |
Dividing by $-3$−3 |

$-3$−3 | $>$> | $-5$−5 |
Simplifying |

The inequality switches from $<$<to $>$>!

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:

**Multiplying**both sides of an inequality by a**negative**number.**Dividing**both sides of an inequality by a**negative**number.

Consider the following statement: $7<10$7<10

Add $6$6 to both sides of the inequality and simplify.

After adding $6$6 to both sides, does the inequality still hold true?

Yes

ANo

B

Consider the following statement: $5<7$5<7

Multiply both sides of the inequality by $2$2 and simplify.

After multiplying both sides by $2$2, does the inequality still hold true?

Yes

ANo

B

Consider the following statement: $6<10$6<10

Multiply both sides of the inequality by $-4$−4 and simplify. Do not change the sign of the inequality.

After multiplying both sides by $-4$−4, does the inequality still hold true?

Yes

ANo

B

Now that we have seen what happens when we perform addition, subtraction, multiplication and division, we can use this knowledge to solve inequalities.

Before jumping in algebraically, it can be helpful to consider some possible solutions and non-solutions. Then we can look at an algebraic strategy.

List at least two values of $x$`x` which satisfy $x+3<4$`x`+3<4 and one which does not.

**Think**: Let's start by picking three values and see if they satisfy the inequality or not. We'll try $-10$−10, $0$0 and $10$10.

**Do:**

Substituting in $-10$−10, we get $-10+3<4$−10+3<4 or $-7<4$−7<4 which is **true**, so $-10$−10 satisfies the inequality

Substituting in $0$0, we get $0+3<4$0+3<4 or $3<4$3<4 which is **true**, so $0$0 satisfies the inequality

Substituting in $10$10, we get $10+3<4$10+3<4 or $13<4$13<4 which is **false**, so $10$10 **does not satisfy** the inequality

So $-10$−10 and $0$0 satisfy the inequality $x+3<4$`x`+3<4, but $10$10 does not. This means that somewhere between $0$0 and $10$10 there is a point where everything below it satisfies the inequality.

**Reflect: **How does knowing some true values help us when finding a solution or graphing?

Solve $x+3<4$`x`+3<4 algebraically and show your work.

**Think: **We can solve this similarly to the equation $x+3=4$`x`+3=4, but we just need to be careful if we are multiplying or dividing by a negative value.

**Do: **

$x+3$x+3 |
$<$< | $4$4 |
Given |

$x+3-3$x+3−3 |
$<$< | $4-3$4−3 |
Subtract $3$3 from both sides |

$x$x |
$<$< | $1$1 |
Simplify both sides |

**Reflect:** How could the two numberlines below help us to visualize this problem?

Solve $-\frac{x}{2}\ge5$−`x`2≥5 algebraically and show your work.

**Think: **We can solve this similarly to the equation $-\frac{x}{2}=5$−`x`2=5, but we just need to be careful when we are multiplying or dividing by a negative value.

**Do: **

$-\frac{x}{2}$−x2 |
$\ge$≥ | $5$5 |
Given |

$-\frac{x}{2}\times\left(-2\right)$−x2×(−2) |
$\le$≤ | $5\times\left(-2\right)$5×(−2) |
Multiply both sides by $-2$−2, we must flip the sign |

$x$x |
$\le$≤ | $-10$−10 |
Simplify both sides |

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

As we have previously seen, we can plot inequalities by using number lines.

We can first solve an inequality and then graph it. This can also help to check our answer.

Solve and graph the solution to $x+3>5$`x`+3>5.

**Think:** We do not need to multiply or divide by a negative, so we end up with a numberline with a hollow (open) endpoint and the ray point to the right.

**Do: **

$x+3$x+3 |
$>$> | $5$5 |
Given |

$x+3-3$x+3−3 |
$>$> | $5-3$5−3 |
Subtract $3$3 from both sides |

$x$x |
$>$> | $2$2 |
Simplify both sides |

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.

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.