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Magazine Chapter 2: Problem 108
Solve $$ 6[4-2(6+3 t)]>5[3(7-t)-4(8+2 t)]-20 $$
Short Answer
Expert verified
t > -\frac{27}{19}
Step by step solution
01
- Simplify Inside Parentheses
Simplify inside the parentheses for both sides of the inequality. Start with the left side: \[ 6[4-2(6+3t)] \rightarrow 6[4-2 \times 6 - 2 \times 3t] \rightarrow 6[4-12-6t] \rightarrow 6[-8-6t] \]
02
- Simplify Inside Parentheses (Right Side)
Now simplify the right side: \[ 5[3(7-t)-4(8+2t)] - 20 \rightarrow 5[21-3t-32-8t]-20 \rightarrow 5[-11-11t] - 20 \]
03
- Distribute Multiplication
Distribute the multiplication on both sides: Left side: \[ 6[-8-6t] \rightarrow 6 \times -8 + 6 \times -6t \rightarrow -48 - 36t \] Right side: \[ 5[-11-11t]-20 \rightarrow 5 \times -11 + 5 \times -11t - 20 \rightarrow -55 - 55t - 20 \rightarrow -75 - 55t \]
04
- Combine Like Terms
Combine like terms for both sides if necessary (already combined here). Now the inequality looks like this: \[ -48 - 36t > -75 - 55t \]
05
- Move Variable Terms to One Side
Add 55t to both sides to move the variable terms to one side: \[ -48 - 36t + 55t > -75 - 55t + 55t \rightarrow -48 + 19t > -75 \]
06
- Move Constant Terms to One Side
Add 48 to both sides to move the constant terms to one side: \[ -48 + 19t + 48 > -75 + 48 \rightarrow 19t > -27 \]
07
- Solve for t
Divide both sides by 19 to solve for the variable t: \[ t > -\frac{27}{19} \]
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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 combinations of variables, constants, and mathematical operations (like addition, subtraction, multiplication, and division). They don't include equality signs or inequalities鈥攖hat's what turns an expression into an equation or inequality.
In the given problem, we have expressions inside the parentheses which need to be simplified first. For instance, on the left side, we have the expression inside 6[4 - 2(6 + 3t)]. Here, we need to perform operations inside each set of parentheses while respecting the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
This exercise demonstrates how to handle nested algebraic expressions and prepare them for further simplification.
In the given problem, we have expressions inside the parentheses which need to be simplified first. For instance, on the left side, we have the expression inside 6[4 - 2(6 + 3t)]. Here, we need to perform operations inside each set of parentheses while respecting the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
This exercise demonstrates how to handle nested algebraic expressions and prepare them for further simplification.
Distributive Property
The distributive property is a crucial algebraic rule which states that multiplying a sum or difference by a number is the same as multiplying each term by the number and then adding or subtracting the results. Mathematically, it can be expressed as:
\[ a(b+c) = ab + ac \]
For the given problem, we use the distributive property to simplify the terms:
On the left side: \[ 6[4-2(6+3t)] 鈫� -48 - 36t \]
On the right side: \[ 5[3(7-t)-4(8+2t)] - 20 鈫� -75 - 55t \]
Thus, the distributive property helps break down complex expressions into simpler ones, making it easier to solve the inequality.
\[ a(b+c) = ab + ac \]
For the given problem, we use the distributive property to simplify the terms:
On the left side: \[ 6[4-2(6+3t)] 鈫� -48 - 36t \]
On the right side: \[ 5[3(7-t)-4(8+2t)] - 20 鈫� -75 - 55t \]
Thus, the distributive property helps break down complex expressions into simpler ones, making it easier to solve the inequality.
Inequality Solving Steps
Solving an inequality involves several systematic steps. Similar to solving equations, we isolate the variable on one side of the inequality. Let's break down the process:
- Simplify both sides of the inequality.
- Distribute any multiplication over addition or subtraction (if necessary).
- Combine like terms on both sides.
- Move all terms containing the variable to one side.
- Move all constant terms to the other side.
- Solve for the variable by performing the necessary arithmetic operations.
Combining Like Terms
Combining like terms is an essential step in simplifying algebraic expressions and inequalities. Like terms are terms that have the same variable raised to the same power.
For example, in the expression -48 - 36t > -75 - 55t, the terms -36t and -55t are like terms because they both contain the variable t raised to the same power.
To isolate the variable, we combined like terms by moving all t terms to one side of the inequality:
Combining like terms throughout these steps helps us simplify the inequality, making it easier to isolate and solve for the variable.
For example, in the expression -48 - 36t > -75 - 55t, the terms -36t and -55t are like terms because they both contain the variable t raised to the same power.
To isolate the variable, we combined like terms by moving all t terms to one side of the inequality:
- Add 55t to both sides: dash; 48 - 36t + 55t > -75 - 55t + 55t 鈫� -48 + 19t > -75
- Then add 48 to both sides: dash; -48 + 19t + 48 > -75 + 48 鈫� 19t > -27
Combining like terms throughout these steps helps us simplify the inequality, making it easier to isolate and solve for the variable.
