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Compute the discriminant. Then determine the number and type of solutions for the given equation. x2 - 2x + 1 = 0
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 2, Problema 79
In Exercises 59–94, solve each absolute value inequality. 3|x - 1| + 2 ≥ 8
Guida verificata passo dopo passo1
Start by isolating the absolute value expression. Subtract 2 from both sides of the inequality: 3\|x - 1\| + 2 \(\geq\) 8 becomes 3\|x - 1\| \(\geq\) 6.
Next, divide both sides of the inequality by 3 to solve for the absolute value: \|x - 1\| \(\geq\) 2.
Recall that the inequality \|A\| \(\geq\) B (where B > 0) means A \(\leq\) -B or A \(\geq\) B. Apply this to \|x - 1\| \(\geq\) 2, giving two inequalities: x - 1 \(\leq\) -2 or x - 1 \(\geq\) 2.
Solve each inequality separately: For x - 1 \(\leq\) -2, add 1 to both sides to get x \(\leq\) -1. For x - 1 \(\geq\) 2, add 1 to both sides to get x \(\geq\) 3.
Combine the solutions to write the final solution set: x \(\leq\) -1 or x \(\geq\) 3.
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Absolute Value Inequalities
An absolute value inequality involves expressions with absolute value symbols, which represent the distance from zero on the number line. To solve these, you often split the inequality into two separate cases based on the definition |A| ≥ B meaning A ≤ -B or A ≥ B, provided B is non-negative.
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06:07
Linear Inequalities
Isolating the Absolute Value Expression
Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves algebraic manipulation such as subtracting or dividing terms, which simplifies the problem and allows you to apply the definition of absolute value inequalities correctly.
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05:09Introduction to Algebraic Expressions
Solving Linear Inequalities
After splitting the absolute value inequality into two linear inequalities, solve each inequality separately using standard methods. This includes adding, subtracting, multiplying, or dividing both sides by constants, while remembering to reverse the inequality sign when multiplying or dividing by a negative number.
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06:07Linear Inequalities
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