GMAT Quant Chapter 5: Inequalities & Absolute Values Flashcards
When we divide or multiply an inequality by a negative sign, what happens?
The inequality sign flips
What must we do before adding two separate inequalities?
Both inequality signs must be facing the same direction
We never perform an operation to an inequality if we don’t know what?
The sign of the unknown.
Never multiply or divide an inequality unless the sign of the variable is known
What is a compound inequality?
A 3 part inequality where an unknown is defined in both directions
When manipulating a compound inequality, how do we maintain the original inequality?
Apply the operation to every individual part of the compound inequality.
What happens when we divide a compound inequality by a negative number?
We must reverse both of the inequality signs.
Which method of solving algebraic equations can we use to solve inequalities and in what order must we perform the operation?
Substitution.
isolate a variable in a separate equation and substitute the result into the inequality.
We cannot isolate a variable in an inequality and substitute that into an equation.
Which part of a compound inequality must we substitute an unknown variable equation into?
It does not matter as long as all manipulations afterwards are applied to all parts of the compound inequality
How should we solve when we are given lots of independent inequalities?
e.g. a > x x < y y < c
Draw a number line and visually see how they relate to each other.
How do we solve inequalities with squared variables?
Solve them like an equation but realise that the root of the squared number produces absolute values so we will often get two solutions.
What are the two solutions when x^2 > B and B > 0
X > Root B
or
X < - Root B
What are the two solutions when x^2 < B, and B > 0?
- Root B < x < Root B
How should we find the minimum and maximum values when we are given two compound inequalities?
evaluate the limits of the 4 inequalities given the smallest will be the minimum and the largest will be the maximum
How many times do we need to solve an equation that has absolute value bars?
Solve the equation twice. Once the absolute value bars are positive and once when they are negative.
How do we solve an equation that absolute value bars only around part of the expression?
Isolate the absolute value bars and put everything else together on the other side.
Then solve the absolute value bars twice.
What do we know if two absolute values are equal?
The expressions within the absolute value bars are either equal or opposite.
what do we know about | a + b | ?
| a + b | < or = |a| + |b|
a + b | < = |a| + |b|
What do we know when | a + b | = |a| + |b|?
I. One or both quantities is 0
II. Both quantities are of the same sign
What do we know about| a - b| ?
|a - b| > or = |a| - |b|
What do we know when |a - b| = |a| - |b|?
Either of the following two things is true:
I. The second quantity is 0
II. Both quantities have the same sign & |a| > or = |b|
How should we mentally rephrase inequalities that contain absolute values?
We are looking for all values of x that are greater or lesser than a certain number away from 0
If |x| > b, does that mean x > b?
Not necessarily.
x > b
or
x < b
Solving which type of inequalities is similar to solving equations that have absolute values?
When we have to square root x^2 because the solution can be either positive or negative.
What is a fundamental truth about absolute values?
How can this fundamental truth stop us from being caught out?
Absolute values are non-negative.
If an absolute value of an expression is set equal to a negative number, there will be no solution to that equation
What must you EVERY SINGLE TIME after solving absolute value equations with multiple solutions?
Check that all of the solutions satisfy the original equation. Often times they don’t and if you don’t remove that incorrect solution you will get the question wrong.
When we take the square root of a binomial ( |a+b|) how many solutions are we likely to get?
- one negative and one positive.
What equation trap frequently occurs in absolute value and inequality problems?
Assuming that the value of a variable cannot be 0.
If the variable can be 0 then we cannot multiply or divide both sides of the equation because it will ruin the solution.
