True or False? A region on a graph can only be bounded by straight lines.

False. The inequalities can be quadratic as well as linear, so a curve can form one of the boundaries.

A shaded region is given on a graph. What do you need before you can write down its inequalities?

The equation of each boundary , worked out from where each line or curve crosses the axes. Only once you have those equations can you decide which way each inequality points.

A LevelMathsEdexcelPureFlashcardsAlgebra & FunctionsInequalities

Inequalities (Edexcel A Level Maths: Pure): Flashcards

Exam code: 9MA0

1/20

0Still learning

Know0

FrontLinear Inequalities
How do you solve a linear inequality?

Was this flashcard helpful?

Cards in this collection (20)

How do you solve a linear inequality?
Treat it exactly like an equation and rearrange to get $x$ on its own.
The one thing to watch is the sign of whatever you multiply or divide by.
On a number line diagram, a $______$ circle shows that the end value is included, and a $______$ circle shows that it is not.
On a number line diagram, a filled-in circle shows that the end value is included, and an empty circle shows that it is not.
So $\leq$ and $\geq$ take the filled circle, while $<$ and $>$ take the empty one.
What is the difference between square and round brackets in interval notation?
Square brackets include the end value and round brackets exclude it.
So $(4, 8]$ means $4 < x \leq 8$ .
Infinity always takes a round bracket, because it is never actually reached.
True or False?
$[- 4, 9)$ and $\{x : x \geq - 4\} \cap \{x : x < 9\}$ describe the same values.
True.
Both say $- 4 \leq x < 9$ , with the $- 4$ included and the $9$ excluded.
Interval notation and set notation are simply two ways of writing the same range.
How do you write $x > 9$ in set notation?
As $\{x : x > 9\}$ , read as "the set of values of $x$ such that $x$ is greater than $9$ ".
A range with two ends joins two such sets with $\cap$ .
Why should you avoid multiplying or dividing an inequality by a negative number?
Because it reverses the inequality sign, and forgetting to reverse it is an easy mistake to make.
Rearranging so the $x$ term is positive avoids the problem: $8 - 3 x \geq 5 x - 4$ becomes $12 \geq 8 x$ , so $x \leq \frac{3}{2}$ .
What is the first thing to do when solving a quadratic inequality?
Rearrange it so that one side is zero and the $x^{2}$ term is positive.
A positive $x^{2}$ term makes the sketch a $\cup$ shape, which is far easier to read a solution from.
Rearrange $5 - 5 x^{2} \leq 7 + 4 x - 8 x^{2}$ so that the $x^{2}$ term is positive:
$______x^{2} -______x -______\leq 0$
$3 x^{2} - 4 x - 2 \leq 0$
Collecting everything on the side that leaves $+ 3 x^{2}$ means the inequality sign never has to be reversed.
A quadratic inequality has been rearranged so the $x^{2}$ term is positive, with roots $x_{1} < x_{2}$ . Where are the solutions?
It depends which way the inequality points:
- greater than zero: outside the roots, so $x < x_{1}$ or $x > x_{2}$
- less than zero: between the roots, so $x_{1} < x < x_{2}$
True or False?
You can multiply both sides of an inequality by $x$ to clear a fraction.
False.
If $x$ could be negative the sign would need reversing, and you do not know its sign in advance.
Multiplying by $x^{2}$ keeps the sign safe but can introduce extra values that do not actually work.
Why is a sketch essential when solving a quadratic inequality?
Because the roots on their own do not tell you which side of them the solution lies.
The sketch shows where the curve sits above or below the $x$ -axis, which is what the inequality is actually asking about.
How do you write the solution $x < - 1$ or $x > 2$ in set notation?
As $\{x : x < - 1\} \cup \{x : x > 2\}$ .
The union symbol is used because the solution is two separate ranges rather than one continuous one.
Do $8 > x > 2$ and $2 < x < 8$ mean the same thing?
Yes, both say that $x$ lies between $2$ and $8$ .
The second is the conventional way of writing it, with the smaller value on the left, and it is much easier to read.
Where do quadratic inequalities come from in discriminant questions?
Setting $b^{2} - 4 a c$ greater than, equal to or less than zero gives an inequality in the unknown constant.
Solving it is then an ordinary quadratic inequality: find the critical values, sketch, and read off the region.
How can you recognise an inequality whose solution is a region on a graph?
It has two variables, $x$ and $y$ , rather than one.
The solution is then an area of the graph rather than a range on a number line.
When drawing inequalities on a graph, use a $______$ line for $<$ or $>$ , and a $______$ line for $\leq$ or $\geq$ .
When drawing inequalities on a graph, use a dotted line for $<$ or $>$ , and a solid line for $\leq$ or $\geq$ .
The solid line shows that the boundary itself is part of the region.
How do you decide which side of a line satisfies an inequality?
Test a point on each side, by substituting its coordinates into the inequality.
The origin is usually easiest: $(0, 0)$ satisfies $y < x^{2} + 1$ , so the side containing the origin is the one you want.
True or False?
A region on a graph can only be bounded by straight lines.
False.
The inequalities can be quadratic as well as linear, so a curve can form one of the boundaries.
When shading a region on a graph, which area do you shade, and how is the answer shown?
Shade the unwanted areas, then label the region left unshaded, usually $R$ .
Some questions ask instead for the wanted region to be shaded, so check the wording before you start.
A shaded region is given on a graph. What do you need before you can write down its inequalities?
The equation of each boundary, worked out from where each line or curve crosses the axes.
Only once you have those equations can you decide which way each inequality points.

Sign up to unlock flashcards

By signing up you agree to our Terms and Privacy Policy

Previous:Simultaneous EquationsNext:Polynomials