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Chapter 2: Problem 76

Solve using the addition and multiplication principles. $$ -5<9 x+8-8 x $$

Short Answer

Expert verified
The solution is $$-5 < x$$ or in interval notation:$$(-5, \infty)$$.

Step by step solution

01

Simplify the expression

Combine like terms on the right-hand side of the inequality. Notice that we have terms involving x: \(9x - 8x\).
02

Combine like terms

Combine \(9x\) and \(-8x\) to get \(x\). The inequality simplifies to: $$ -5 < x$$
03

Isolate the variable

The inequality is already simplified and the variable x is isolated. Thus, the solution to the inequality is: $$ -5 < x$$
04

Write the solution

Express the solution in interval notation: $$(-5, \infty)$$

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Key Concepts

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

Inequality Simplification
Simplifying an inequality involves reducing it to its simplest form. This process often includes combining like terms and eliminating unnecessary elements. In the given problem, we start with the inequality:
-5 < 9x + 8 - 8x
Our goal is to simplify the right-hand side by combining like terms. Here, the like terms involving x are 9x and -8x. Adding these together results in a simplified term:
x
Now, the inequality looks cleaner and is easier to work with:
-5 < x
Addition Principle
The addition principle is used to add (or subtract) the same amount from both sides of an equation or inequality to maintain its balance. In our simplified inequality:
-5 < x
You would typically use the addition principle if you had terms on both sides of the inequality that need balancing. For instance, if our inequality were something like -5 + 3 < x - 2, we could use addition or subtraction to isolate x. But in our case, no further addition or subtraction is needed because x is already isolated:
-5 < x
Multiplication Principle
The multiplication principle helps solve inequalities by multiplying (or dividing) both sides by the same non-zero number. You have to be careful with inequalities, as reversing the inequality sign is necessary when multiplying or dividing by a negative number.
For our inequality:
-5 < x
We don't need to apply this principle as x is already isolated and no multiplication or division is required. However, if your problem had been something like -5 > 2x, you would divide both sides by 2 to isolate x, resulting in -2.5 > x. Remember, always flip the inequality sign when multiplying or dividing by a negative number.
Interval Notation
Interval notation is a way to express solutions to inequalities succinctly. It indicates the range of values that satisfy the inequality. For the final simplified inequality:
-5 < x
We can express this in interval notation as (-5, 鈭�).
Here's how interval notation works:
  • A parenthesis ( or ) means that the endpoint is not included in the interval.
  • An infinity symbol 鈭� is used to signify that the interval extends indefinitely in the positive or negative direction.

So (-5, 鈭�) means that x can be any number greater than -5 but not including -5 itself, and it can go up to infinity. This notation makes it easy to understand and communicate the range of numbers x can take.

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Most popular questions from this chapter

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