Is the lowest common denominator the only way to clear fractions from an equation?

No, you can multiply by one denominator first and then by the other, and both routes reach the same answer. Using the lowest common denominator is quicker, because it clears every fraction in a single step.

National 5MathsSQAFlashcardsAlgebraic SkillsLinear Equations & Inequations

Linear Equations & Inequations (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontLinear Equations
Define a linear equation.

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Cards in this collection (16)

Define a linear equation.
A linear equation is one that can be written in the form $a x + b = c$ , where $a$ , $b$ and $c$ are numbers.
The highest power of $x$ is $1$ , so there are no terms such as $x^{2}$ .
Complete the solution by filling in the two missing values.
$3 x - 7 = 35 \Rightarrow 3 x =______\Rightarrow x =______$
The completed solution is:
$3 x - 7 = 35 \Rightarrow 3 x = 42 \Rightarrow x = 14$
Each step does the same thing to both sides, adding $7$ and then dividing by $3$ .
What must be done first when a linear equation contains brackets?
Expand the brackets before anything else, so $2 (x - 3) = 10$ becomes $2 x - 6 = 10$ .
Adding $6$ and then dividing by $2$ gives $x = 8$ .
With $x$ terms on both sides which one should you remove?
Remove the smallest $x$ term, which is the most negative one if either has a minus sign, because that avoids working with negatives.
In $3 - 5 x = 6 (x - 5)$ that means adding $5 x$ to both sides rather than subtracting $6 x$ .
How do you deal with fractions in a linear equation?
Multiply both sides by the lowest common denominator, which clears every fraction at once.
For $\frac{x}{5} + 4 = \frac{9}{2}$ the lowest common denominator is $10$ , and multiplying through gives $2 x + 40 = 45$ .
True or False?
Multiplying an equation by the lowest common denominator means multiplying every term, including any that are not fractions.
True.
Both sides have to stay balanced, so a term that is not a fraction is multiplied just like the ones that are.
Multiplying $\frac{4 x + 1}{2} - 7 = \frac{5 x}{3}$ by $6$ turns the $- 7$ into $- 42$ , giving $3 (4 x + 1) - 42 = 10 x$ .
How is $\frac{4}{x - 2} = 3$ solved when the unknown is in the denominator?
Multiply both sides by that denominator, which gives $4 = 3 (x - 2)$ .
Expanding and solving then gives $4 = 3 x - 6$ , so $x = \frac{10}{3}$ .
Is the lowest common denominator the only way to clear fractions from an equation?
No, you can multiply by one denominator first and then by the other, and both routes reach the same answer.
Using the lowest common denominator is quicker, because it clears every fraction in a single step.
Define an inequation.
An inequation, also called an inequality, states that one quantity is greater than or less than another.
The signs $>$ , $<$ , $\geq$ and $\leq$ take the place of an equals sign.
What values can $x$ take when $x > 5$ is given?
Any value greater than $5$ at all, not just whole numbers, so $5.25$ and $\sqrt{26} = 5.0990195 \dots$ both work.
The sign $>$ is strict, so $5$ itself is excluded, whereas $x \geq 5$ would include it.
True or False?
Solving an inequation follows exactly the same rules as solving an equation.
False.
Almost all of it is the same, but multiplying or dividing both sides by a negative number reverses the direction of the sign.
Starting from $1 < 2$ and multiplying both sides by $- 1$ gives $- 1 > - 2$ .
Complete the solution by filling in the missing sign.
$- 5 x < 15 \Rightarrow x______- 3$
The completed solution is:
$- 5 x < 15 \Rightarrow x > - 3$
Dividing both sides by $- 5$ reverses the direction of the inequation.
Why is multiplying an inequation by a variable such as $x$ never allowed?
A variable can be positive or negative, so there is no way of knowing whether the direction of the sign should be reversed.
Multiplying $x < 3$ by $x$ would give $x^{2} < 3 x$ , which is not true when $x$ is negative.
How can you solve an inequation without ever dividing by a negative number?
Collect the variable terms on whichever side leaves the coefficient positive, even if that means the unknown ends up on the right-hand side.
Solving $4 x - 5 < 9 x + 10$ that way gives $- 15 < 5 x$ and then $- 3 < x$ .
True or False?
$- 3 < x$ and $x > - 3$ say exactly the same thing.
True.
Reading $- 3 < x$ from the right says that $x$ is greater than $- 3$ , which is what $x > - 3$ says.
Flipping an inequation over reverses the sign along with the two sides, so the meaning is unchanged.
How do you list the integers that satisfy $- 4 \leq x < 2$ correctly?
Check each end separately to see whether it is included, then count through every whole number in between.
Here $- 4$ is included and $2$ is not, so the list is $- 4, - 3, - 2, - 1, 0, 1$ .
Remember that zero and negative whole numbers count as integers.

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