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Chapter 1: Problem 85

Evaluate \(x^{2}-x\) for the value of \(x\) satisfying \(4(x-2)+2=4 x-2(2-x)\)

Short Answer

Expert verified
Evaluating the expression \(x^{2}-x\) for the value of \(x\) satisfying \(4(x-2)+2=4x-2(2-x)\) results in \(0\).

Step by step solution

01

Simplify both sides of the equation

Expand the equations on both sides to simplify them. On the left side, we have \(4x-8+2\), which simplifies to \(4x-6\). On the right side, expanding yields \(4x - 2(2) + 2x\), which simplifies to \(6x - 4\). Therefore, our equation becomes \(4x - 6 = 6x - 4\).
02

Solve for x

To solve for \(x\), we can subtract \(4x\) from both sides which gives \(2x = 2\). Then divide both sides by 2 to get \(x = 1\).
03

Substitute the value of x into the expression

Now we substitute \(x = 1\) into the given expression \(x^{2}-x\). This gives us \(1^{2} - 1 = 0\).

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Key Concepts

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

Solving Equations
When you have an equation, your main goal is to find out what value or values make it true. This exercise involves balancing one side of an equation with the other. To solve it, start by looking at the equation: \[4(x-2)+2=4x-2(2-x)\]Begin by simplifying both sides. You'll perform operations such as distributing and combining like terms. For instance:
  • The left side simplifies to \(4x - 6\).
  • The right side simplifies to \(6x - 4\).
Now you have:\[4x - 6 = 6x - 4\]To isolate \(x\), subtract \(4x\) from both sides, which gives the equation: \(2x = 2\). Divide both sides by \(2\) to find \(x = 1\). This process is key for solving equations as it balances both sides while isolating the variable.
Substitution Method
The substitution method is handy once you've solved an equation and found a value for the variable. With \(x = 1\), you can replace \(x\) in the expression you want to evaluate. This expression given in the problem is \(x^2 - x\). To apply the substitution method:
  • Replace every instance of \(x\) with \(1\).
  • Calculate the new expression: \(1^2 - 1 = 0\).
By substituting \(x\) with its value, we can quickly find the solution to the expression. This step ensures that we're using the right value to get an accurate result, reflecting the solution of the initial equation.
Simplification
Simplification makes an equation or expression easier to work with by reducing it to a more basic form. It's crucial in both solving equations and evaluating expressions. As illustrated in this exercise:To simplify the original equation, you performed these steps:
  • Distribute any multiplications across additions or subtractions, such as \(4(x-2)\) becoming \(4x - 8\).
  • Combine like terms, such as \(-8 + 2\) becoming \(-6\).
This simplification helps create a clearer path to finding \(x\). Once you know \(x\), simplify the expression \(x^2 - x\) by replacing \(x\) and calculating into \(1^2 - 1\) which gives \(0\). Each simplification step sharpens accuracy, making math challenges easier and more efficient to solve.

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Most popular questions from this chapter

Solve absolute value inequality. \(5>|4-x|\)

Solve absolute value inequality. \(-2|5-x|<-6\)

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

When 3 times a number is subtracted from \(4,\) the absolute value of the difference is at least \(5 .\) Use interval notation to express the set of all numbers that satisfy this condition.
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