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Chapter 3: Problem 20

Determine whether the given value of the variable satisfies the inequality. $$-5<8-2(x+3) \leq 0 ; \quad x=4$$

Short Answer

Expert verified
The value x=4 does not satisfy the inequality.

Step by step solution

01

Substitute the given value

Substitute the given value of x into the inequality. The given value is x = 4, so the inequality becomes \[-5 < 8 - 2(4 + 3) \leq 0\].
02

Simplify inside the parentheses

Calculate the expression inside the parentheses: \[4 + 3 = 7\]. So the inequality is now \[-5 < 8 - 2(7) \leq 0\].
03

Perform the multiplication

Multiply -2 by 7: \[2(7) = 14\]. Hence the inequality becomes \[-5 < 8 - 14 \leq 0\].
04

Simplify the inequality

Simplify the expression: \[8 - 14 = -6\]. So the final inequality is \[-5 < -6 \leq 0\].
05

Check the inequality

Check if the simplified inequality holds true. Check each part:1. \[-5 < -6\]: This is false because -6 is not greater than -5.2. \[-6 \leq 0\]: This is true because -6 is less than 0.Since one part of the compound inequality is false, the entire inequality is not satisfied.

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Key Concepts

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

algebraic expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. For example, in our given inequality problem \[-5<8-2(x+3) \leq 0\], the expression \[-2(x+3)\] is an algebraic expression.
When solving inequalities, we often need to work with these expressions, simplifying them step-by-step.
This involves performing operations inside parentheses first.
We then proceed with multiplication or division, and finally, any addition or subtraction required to simplify the inequality expression.
inequality verification
Inequality verification involves checking whether a particular value satisfies the given inequality.
To verify the inequality \[-5<8-2(x+3) \leq 0\] for the given value x = 4, we substitute 4 into the inequality: \[-5 < 8 - 2(4 + 3) \leq 0\].
Once substituted, we simplify the expression.
If for any part of the compound inequality, like \[-5 < -6\], the statement is false, the whole inequality isn't satisfied.
Thus, verification is an essential step to check if our simplified expression stands true within the constraints of the inequality.
compound inequalities
Compound inequalities consist of two separate inequalities that are joined together by either 'and' or 'or'.
In our problem, we have the compound inequality \[-5 < 8 - 2(x + 3) \leq 0\], where both parts \[-5 < -6\] and \[-6 \leq 0\] need to be satisfied for the overall inequality to hold.
The key point in compound inequalities is to simplify each part individually and then check their combined validity.
This means handling the algebraic expressions correctly and ensuring that each condition after simplification can logically present a true statement.
substitution method
The substitution method in solving inequalities involves replacing variables with their given values.
For instance, substituting x = 4 in \[-5 < 8 - 2(x + 3) \leq 0\] helps transform the inequality into a simpler numerical form.
We replace x with 4, which changes our inequality into \[-5 < 8 - 2(4 + 3) \leq 0\].
Next, we simplify inside the parentheses, perform the multiplications, and simplify the entire expression.
This method simplifies the evaluation of the compound inequality and is useful for checking the validity of specific variable values within the equation.

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Most popular questions from this chapter

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