The solution set is {x | x < -3 or x > 5}.
Solve The Compound Inequality 4X 15
To solve the compound inequality 4x + 15, it is necessary to evaluate the expressions within the parentheses and then compare them to determine whether they are less than or greater than each other. First, isolate the right hand side of the equation. This can be done by subtracting 15 on both sides of the equation, leaving 4x as the only term on one side. Next, factor out 4 on both sides of the equation to obtain x as a linear expression. Finally, compare x to -3.5 to determine whether x is greater than or less than -3.5. The result will provide us with the answer to our inequality: 4x + 15 < -3.5 or 4x + 15 > -3.5.
Define Compound Inequalities
A compound inequality is an inequality statement that combines two or more simple inequalities into one statement. The two inequalities can be joined by the words and or or, as well as various symbols such as <, >, , . For example, the compound inequality 4x + 15 < 24 and x > 2 can be written as 4x + 15 < 24 x > 2. As with simple inequalities, the solution set of a compound inequality is the set of all values that make the overall statement true.
Step-by-Step Simplification Procedure
The first step in solving a compound inequality is to simplify it by combining like terms on each side of the equation and using properties of inequalities such as transitive property. In this case, we have 4x + 15 < 24 and x > 2. To simplify this expression, we begin by combining like terms on both sides of each inequality:
4x + 15 < 24 4x < 9
and
x > 2 x > 2
Next, we use transitive property to see if any of these two equations can be combined into one: 4x < 9 and x > 2 4x + 15 < 24. In this case, they cannot be combined so we are left with two separate equations which we can now solve for x.
Solve for x
To solve for x in a compound inequality, we must first identify which side of each equation contains the variable and then use various methods to solve it depending on what type of equation it is. For example, if one side contains only constants (numbers without any variables) then we can use algebraic methods such as factoring or completing the square to solve it. On the other hand, if both sides contain variables then we can use graphing methods such as plotting points or constructing a graph to identify solutions. In our case, the first equation has only constants on one side so we will use algebraic methods to solve for x:
4x < 9 4x +4 = 9 4(x+1)=9 x+1=9/4 x= 8/4 - 1 x=2
For our second equation we have both constants and variables so we will use graphing methods to find solutions: Plotting points or constructing a graph will help us identify all solutions that make this equation true. We begin by plotting points on a coordinate plane where each point is represented by an ordered pair (a number on one axis and its corresponding number on another axis). For our equation above (x > 2), every point that lies above the line y = 2 will represent solutions that make this equation true because they all have an ‘x’ coordinate greater than ‘2’. After plotting several points above y = 2 (such as (3;4), (4;5) etc.)we see that any point located above this line represents a solution for our equation which means that our solutions are all numbers greater than ‘2’.
Identifying Solutions
Once we have identified all possible solutions for our compound inequality, it’s important to verify them by substituting them back into our original equations to make sure they are correct. This step is important because even though our graphical representation may show us several possible solutions there might still be some incorrect ones among them. To verify them, substitute each solution back into both original equations and see if they make them true: Our solution set consists of all numbers greater than ‘2’ so let’s substitute 3 into both equations: 4(3)+15=27<24 is TRUE 3>2 is TRUE Both statements are TRUE which means that 3 is indeed a valid solution for our compound inequality! Similarly verify other potential solutions you may have identified using graphing methods until you find all valid solutions in your solution set.
Verifying Solutions
Once you have identified your solution set using either graphical methods or algebraic ones it’s important to verify them before considering your problem solved! This step ensures accuracy since sometimes even though graphical representation may show us several possible solutions some of these might still be incorrect! To do this simply substitute each solution back into both original equations and check if they make them true: Our solution set consists of all numbers greater than ‘2’ so let’s substitute 3 into both equations: 4(3)+15=27<24 is TRUE 3>2 is TRUE Both statements are TRUE which means that 3 is indeed a valid solution for our compound inequality! Similarly verify other potential solutions you may have identified using graphing methods until you find all valid solutions in your solution set.
Graphical Representation Of The Answer
The final step in solving a compound inequality problem is to represent it graphically – typically done on a coordinate plane where each point represents an ordered pair (a number on one axis and its corresponding number on another axis). This helps us visualize what sets of values satisfy the given conditions which makes understanding much easier! In this case, since our answer was “all numbers greater than ‘2’”, we would plot several points above y = 2 (such as (3;4), (4;5) etc.)to show how every point located above this line represents a valid solution for our problem!
Plotting the Solution on a Number Line
When trying to solve a compound inequality such as 4x 15, it is important to plot the solution on a number line. This will help to visualize the solution and see how it is affected by different inequalities. The first step in plotting the solution is to find out what values of x satisfy the inequality. To do this, we need to solve 4x 15 for x. We can do this by adding 15 to both sides of the equation, giving us 4x = 15 + 15 or 4x = 30. We then divide both sides by 4, giving us x = 7.5. This means that any value of x that is greater than 7.5 will satisfy the inequality, while any value of x that is less than 7.5 will not satisfy it.
We can now plot this value on a number line, as shown below:
As we can see from this graph, any value of x that lies on or to the right of point 7.5 (marked with a black dot) will satisfy the inequality, while any value of x that lies on or to the left of point 7.5 will not satisfy it.
Analyzing The Graph
Now that we have plotted our solution on a number line, we can analyze it further. In this case, we can see that all values of x between 7 and 8 (including 7 and 8) will satisfy our compound inequality 4x 15; this makes sense since these are all greater than our solution point 7.5 which we found earlier when solving for x in our equation 4x 15 = 0 . We can also see that all values of x which are less than 7 (including -7 and -8) will not satisfy our compound inequality; again this makes sense since these are all less than our solution point 7.5 which we found earlier when solving for x in our equation 4x 15 = 0 .
Conceptual Understanding Of The Solution
It is important to have an understanding not just of how the solution looks visually on a number line but also what it means conceptually when solving an inequality such as 4x -15= 0 . In this case, we are looking at all possible values of X which are greater than or equal to our solution point 7.5; these values form what is known as an ‘interval’, and they represent all possible solutions which would satisfy our compound inequality 4x 15=0 . To put it another way, if X is any real number greater than or equal to 7.5 then it satisfies the equation; if X is any real number less than 7.5 then it does not satisfy it .
Interpreting The Solution Inequality
Now that we have understood what interval notation looks like visually on a number line and conceptually behind-the-scenes when solving an inequality such as 4x -15= 0 , let’s take a look at how we might interpret this information in written form using standard mathematical notation for inequalities:
4x 15 0
This means For all real numbers X where X 7 . 5 , then 4X -15 0
In other words: Any value of X which is greater than or equal to point 7 . 5 satisfies our compound inequality 4X -15 0 ; any value of X which is less than point 7 . 5 does not satisfy it .
Interval Notation Representation
Now that we have looked at how to interpret an interval notation expression such as 4X -15 0 in English language terms let’s take a look at how we might represent this same information using interval notation: [7 . 5 , )
This expression simply tells us that all numbers greater than or equal to point 7 . 5 (7 . 5 included) form an open interval denoted by square brackets ( [ ) and parentheses ( ) ). Any value outside this open interval (i . e anything less than point7 . 5 ) does not form part of the interval and therefore does not satisfy the given expression 4X -15 0
FAQ & Answers
Q: What is a compound inequality?
A: A compound inequality is a statement that contains two inequalities that are connected by the words “and” or “or”. It can be used to describe the relationship between two different values.
Q: How do I simplify a compound inequality?
A: To simplify a compound inequality, first isolate the variable from each of the two inequalities. Next, combine the inequalities by using the appropriate connecting words (and or or). Finally, solve for the variable.
Q: How do I solve for x in a compound inequality?
A: To solve for x in a compound inequality, first isolate the variable from each of the two inequalities. Next, combine the inequalities by using the appropriate connecting words (and or or). Finally, solve for x by evaluating each side of the equation and identifying which solutions are valid.
Q: How do I graphically represent an answer to a compound inequality?
A: To graphically represent an answer to a compound inequality, plot each solution on a number line and draw lines between them to show which solutions are valid. It is also important to analyze what this graph is saying about your solution.
Q: What does it mean to interpret my solution as an inequality?
A: Interpreting your solution as an inequality means understanding that it describes how two values relate to each other. For example, if your solution is x > 4, this means that any value of x that is greater than 4 will satisfy this equation. You can also represent this solution in interval notation, which shows all of the possible values of x that satisfy this equation.
The solution to the compound inequality 4x 15 is x 3.75. This means that any value of x that is greater than or equal to 3.75 will satisfy the inequality, while values that are less than 3.75 will not satisfy the inequality.
