Solve Multi Step Inequalities in Math
You will see various types of questions on how to solve multi step inequalities on your algebra test.
This post provides an overview of linear, absolute value, and quadratic inequalities.
Solve Multi Step Inequalities – Quiz
Instructions: Find all possible values of x for the inequalities provided. You may want to study the examples below the quiz first.
| [WpProQuiz 28] |
Solve Multi Step Inequalities – Linear Inequality Example
The way we go about solving linear inequalities is very similar to the process used in solving linear equations.
You need to deal with the integers in the inequality, and then isolate the variable.
Look at the example linear inequality problem below.
Example: Solve for possible values of x.
–3(2x + 5) > 9
Answer: x < –4
STEP 1:
Perform operations on the parentheses.
–3(2x + 5) > 9
–3 × (2x + 5) > 9
(–3 × 2x) + (–3 × 5) > 9
–6x + (–15) > 9
–6x – 15 > 9
STEP 2:
Deal with the integers.
–6x – 15 > 9
–6x – 15 + 15 > 9 + 15
–6x > 9 + 15
–6x > 24
STEP 3:
Isolate x to solve.
–6x > 24
–6x ÷ –6 > 24 ÷ –6
Remember When to Change Signs!
Remember that when you divide or multiply by a negative in an inequality, you need to change the direction of the sign.
–6x ÷ –6 > 24 ÷ –6 We are dividing by a negative number here.
x < 24 ÷ –6 So, now need to use the “less than” symbol.
x < –4
Check Your Answer
You can check your work to be sure that you have the inequality sign pointing in the right direction.
We will use –5 for x in the proof below since it satisfies the statement that x < –4.
–3(2x + 5) > 9
–3[(2 × –5) + 5] > 9
–3[(–10) + 5] > 9
(–3 × –10) + (–3 × 5) > 9
30 – 15 > 9
15 > 9
Solving Inequalities – Absolute Value Example
Absolute value can be thought of as the distance of a number from zero on a number line.
So, absolute value is always a positive number.
Using the absolute value symbol, we can demonstrate as follows: |–5| = 5
If this concept is not familiar to you, please view our post on absolute value equations before continuing with this example.
Be sure to consider both values!
To solve an absolute value inequality in the form |ax + b| > c, you need to solve for the following two inequalities:
ax + b > c
ax + b < –c
Now let’s look at an example.
Example: Solve for x in | 2x – 6| > 8
Answers: x > 7 and x < –1
STEP 1:
Solve for the value of c in the form ax + b > c .
2x – 6 > 8
2x – 6 + 6 > 8 + 6
2x > 8 + 6
2x > 14
2x ÷ 2 > 14 ÷ 2
x > 7
STEP 2:
Solve for the value of –c in the form ax + b < –c .
2x – 6 < –8
2x – 6 + 6 < –8 + 6
2x < –8 + 6
2x < –2
2x ÷ 2 < –2 ÷ 2
x < –1
Absolute Inequality Rule
Be careful not to confuse solving absolute value inequalities with the absolute value inequality rule.
The absolute value inequality rule says that the absolute value of the sum of two numbers is always less than or equal to the sum of the absolute values of the individual numbers.
Using the absolute value symbols | | , we can represent this rule as follows:
| x + y | ≤ |x| + |y|
Absolute Inequality Proof
We can prove that this rule is true, when a = 2 and b = –3, for example:
| x + y | ≤ |x| + |y|
| 2 + –3 | ≤ |2| + |–3|
| 2 – 3 | ≤ 2 + 3
| –1 | ≤ 2 + 3
1 ≤ 2 + 3
1 ≤ 5
Solving Inequalities – Quadratic Inequality Example
You will remember from our post on quadratic forms that a quadratic has a variable that is raised to the power of two.
When working with quadratics, you need to bear in mind that there will usually be more than one possible solution to a quadratic inequality.
Let’s look at an example of how to solve a quadratic inequality.
Example: Solve for possible values of x.
x2 + 2x > 15
Answers: x < –5 and x > 3
STEP 1:
Move the terms to one side of the inequality so that you have just zero on one side.
x2 + 2x > 15
x2 + 2x – 15 > 15 – 15
x2 + 2x – 15 > 0
STEP 2:
Factor the quadratic. If you do not know how to to this, please see our post on factoring quadratics before you finish studying this example.
x2 + 2x – 15 > 0
(x – 3)(x + 5) > 0
STEP 3:
Find possible values for x by making the value inside each set of parentheses equal to zero.
For the first set of parentheses (x – 3), we would need to replace x with 3 to make the value zero (because 3 – 3 = 0).
So, we know for this set of parentheses that x > 3.
For the second set of parentheses (x + 5), we would need to replace x with –5 to make the value zero (because –5 + 5 = 0).
To solve this part of the quadratic, we need to reverse the way the sign points.
So, we know for the second set of parentheses that x < –5.
STEP 4:
Check for possible values of x.
From step 3 above, we know that x > 3.
So, replace x for a value that is greater than 3.
We will use 4 as an example in our proof for this quadratic.
(x – 3)(x + 5) > 0
(4 – 3)(4 + 5) > 0
(4 – 3)(4 + 5) > 0
1 × 9 > 0
9 > 0
Check Your Answer
Now check your other possible answer.
From step 3 above, we also know that x < –5.
So, replace x with a value that is less than –5.
We will use –6 as an example in our proof for the quadratic.
(x – 3)(x + 5) > 0
(–6 – 3)(–6 + 5) > 0
(–6 – 3)(–6 + 5) > 0
–9 × –1 > 0
9 > 0
So, our answers are x < –5 and x > 3.
Solve Multi Step Inequalities – Practical Problems
You may also need to know how to use inequalities to solve practical problems like the one in this section
Example: In the equations below, x represents the cost of a pair of jeans and y represents the cost of a pair of shoes. If we know that x – 3 > 5 and y = x – 3, then the cost of 2 pairs of shoes is greater than which one of the following?
A. x − 2
B. x − 5
C. y + 5
D. 5
E. 10
Look for Common Terms
For problems like this, look to see if both of the equations have any variables or terms in common.
In this problem, both equations contain x − 3.
The cost of one pair of shoes is represented by y, and y is equal to x − 3.
Therefore, we can substitute values from one equation to another.
x − 3 > 5
y > 5
If two pairs of jeans are being purchased, we need to solve for 2y.
y × 2 > 5 × 2
2y > 10
So, the correct answer is E.
Further Exercises
You may also want to view our posts on:
For help with rational inequalities, please see our posts on algebraic fractions.
For more help with graphing inequalities, please see our posts on graphing.
