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Chapter 5: Problem 2

Solve the following linear inequalities. $$ x+4>9 $$

Short Answer

Expert verified
Answer: The solution to the linear inequality is x > 5. The graphical representation is a number line with an open circle at 5 and an arrow pointing to the right, indicating all numbers greater than 5.

Step by step solution

01

Identify the inequality

We are given the linear inequality: $$ x + 4 > 9 $$
02

Isolate the variable

To isolate the variable "x", we must subtract 4 from both sides of the inequality: $$ x + 4 - 4 > 9 - 4 $$ Therefore, we have: $$ x > 5 $$
03

Represent the solution graphically

Now, let's represent the solution to this inequality on a number line. Since x is greater than 5 but not equal to 5, we use an open circle at 5 and an arrow pointing to the right to represent all numbers greater than 5. (-----:-----:------:-----) 3 4 5 6 7 Here, the open circle is placed at the number 5 and the arrow points to the right, indicating all the numbers greater than 5 are part of the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Inequalities
When we talk about linear inequalities, we're referring to mathematical expressions that involve a variable (usually represented as 'x' or 'y') and use inequality symbols such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Unlike linear equations that have a single solution, linear inequalities have a range of solutions, making up what we call a solution set.

For instance, the inequality 'x + 4 > 9' tells us that when you add 4 to a number 'x', it's greater than 9. However, it doesn't give us a single number but instead a set of numbers that fulfill this condition. In essence, we're exploring all the possible values 'x' can take that will satisfy the inequality.
  • These expressions can often be manipulated similarly to equations but require special attention when multiplying or dividing by a negative number, as this will reverse the inequality sign.
Inequality Representation
To truly understand an inequality, it's helpful to visualize what it represents, and that's where inequality representation comes into play. It's a method of showing the range of solutions of an inequality in a graphical format which is usually done on a number line.

For our exercise example 'x > 5', the open circle represents the number 5 to indicate that this number is not included in the solution set - because 'x' is not equal to 5, it's only greater than 5. An arrow extending to the right from this open circle signals that all numbers to the right of 5 are part of the solution set. This means any number greater than 5 is a valid solution for the inequality.
  • It's crucial to remember that open circles denote 'greater than' (>) or 'less than' (<) while closed circles are used for 'greater than or equal to' (≥) or 'less than or equal to' (≤).
Isolating the Variable
The process of isolating the variable is a critical step in solving both equations and inequalities. It involves rearranging the expression so that the variable of interest stands alone on one side of the inequality (or equation) sign, and everything else is on the other side. This is usually accomplished through basic arithmetic operations: adding, subtracting, multiplying, or dividing.

In our exercise, to isolate 'x' from 'x + 4 > 9', we subtract 4 from both sides, giving us 'x > 5'. This simplifies the inequality to clearly display the solution set of 'x'.
  • When isolating the variable, always perform the same operation on both sides of the inequality to maintain the balance.
  • If you ever multiply or divide by a negative number, don't forget to reverse the inequality sign to preserve the truth of the statement.

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Most popular questions from this chapter

For the following problems, solve the linear equations in two variables. $$ -y=0 x+10, \text { if } x=3 $$

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