Solving Quadratic Inequality x^2 - 2x + 1 < x

Math Problem Statement

ou’ve already practiced solving quadratic equations using a variety of techniques, including factoring and using the quadratic formula. There may also be times when you need to solve quadratic inequalities. The process of solving quadratic inequalities is similar to the process of solving quadratic equations, but it requires a slightly different approach at the end.

Normally, once you factor a quadratic equation that is equal to 0, you set the different factors equal to 0. You can’t do this with inequalities. This is because the direction of inequalities changes when a negative number is multiplied or divided across the sign. Because variables can take on either positive or negative values, we can’t separate the factors as we do with an equation.

In this task, you’ll solve quadratic inequalities by testing intervals that are produced by a process similar to that of solving quadratic equations.

Part A Rewrite this inequality so that one side is 0.

x2 − 2x + 1 < x − 1

Solution

To solve the quadratic inequality $x^2 - 2x + 1 < x - 1$ , we first want to rewrite it so that one side is 0.

Step 1: Move all terms to one side of the inequality.

Subtract $x - 1$ from both sides:

$x^2 - 2x + 1 - (x - 1) < 0$

Simplify:

$x^2 - 2x + 1 - x + 1 < 0$

$x^2 - 3x + 2 < 0$

Now, the inequality is in the form $x^2 - 3x + 2 < 0$ , with one side equal to 0.

Would you like to proceed with solving this inequality by finding its solution intervals? Let me know if you need more clarification or a detailed breakdown.

Here are some related questions to help expand your understanding:

How can you factor quadratic expressions like $x^2 - 3x + 2$ ?
What are the key differences between solving quadratic equations and quadratic inequalities?
What is the test interval method for solving quadratic inequalities?
How does the direction of an inequality change when you multiply or divide by a negative number?
Can you explain the concept of critical points in solving quadratic inequalities?

Tip: Always check the sign of the quadratic expression in each interval defined by the roots to determine where the inequality holds.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Inequalities
Inequality Manipulation

Formulas

Theorems

Suitable Grade Level

Grades 9-11

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