Inequalities Maths

$x+2<7$

$x^2+x-20<0$

You know that 4> 2, but if you multiply this inequality by -1

Then you get -4> -2 which is not true

For the inequality to remain true, you need to reverse the symbol, like this:

-4 <-2 ✔ which is true

This is because, in the case of negative numbers, the closer the Number is to zero, the larger it is.

You can see -4 and -2 represented on the Number Line as follows:

Numbers on the number line, Marilú García De Taylor - StudySmarter Originals

1) x - 5> 8 isolate x and combine like terms

x> 8 + 5

x> 13

Using set notation, the solution is {x: x> 13}, which you can read as the set of values of x for which x is greater than 13.

2) 2x + 2 <16 isolate x and combine like terms

2x <16 -2

2x <14

$x<\frac{14}{2}$

x <7

Set notation: {x: x <7}

3) 5 - x <19

- x <19 - 5

- x <14 Remember to change the symbol, as you are dividing by -1

x> -14

Set notation: {x: x> -14}

4) If you need to find the set of values for which two inequalities are true together, you can use a number line to see the solution more CLEARLY.

The solution will be the values that satisfy both Equations at the same time. For example:

Solving linear inequalities using the number line, Marilú García De Taylor - StudySmarter Originals

Set notation: {x: 4 <x <5}

If there is no overlap, then the inequalities are written separately.

Solving linear inequalities using the number line - no overlap, Marilú García De Taylor - StudySmarter Originals

Set notation: {x: x <4} ∪ {x: x> 5}

• Find the set of values of x for which $x^2+x-6>0$

$x^2+x-6=0$ factorise to find the critical values

(x - 2) (x + 3) = 0

The critical values are: x = 2 and x = -3

You can use a table to help you see where the graph will be positive or negative.

You can read the information on the table like this: If x <-3, (x - 2) is negative, (x + 3) is negative, and (x - 2) (x + 3) is positive, and the same for the other columns. The last row (x - 2) (x + 3) tells you where the graph will be positive or negative.

Now you can draw the graph:

Solving quadratic inequalities graph, Marilú García De Taylor - StudySmarter Originals

The solution to $x^2+x-6>0$ are the values of x where the curve is above the x-axis. This happens when x <-3 or x> 2. In set notation: {x: x <-3} ∪ {x: x> 2}

Solving quadratic inequalities graph - curve above the x-axis, Marilú García De Taylor - StudySmarter Originals

• If you want to find the solution for $x^2+x-6<0$, it will be the values of x where the curve is below the x-axis. This happens when -3 <x <2. In set notation: {x: -3 <x <2}

Solving quadratic inequalities graph - curve below the x-axis, Marilú García De Taylor - StudySmarter Originals

Given the Equations y = 3x + 10, and $y=x^2$, find the solution for the inequality$3x+10>x^2$

Make the equations equal to each other to find the points of intersection and the critical values:

$3x+10=x^2$

$x^2-3x-10=0$ factorise to find the critical values

$\left(x+2\right)\left(x-5\right)$

The critical values are x = -2 and x = 5

Substitute the critical values into $y=x^2$ to find the points of intersection :