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In Exercises 59–94, solve each absolute value inequality. |(2x + 2)/4| ≥ 2
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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 75
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| + 3 = 3
Guida verificata passo dopo passo1
Start by isolating the absolute value expression on one side of the equation. Given the equation \(|2x - 1| + 3 = 3\), subtract 3 from both sides to get \(|2x - 1| = 0\).
Recall that the absolute value of an expression equals zero only when the expression inside the absolute value is zero. So, set the inside equal to zero: \(2x - 1 = 0\).
Solve the linear equation \(2x - 1 = 0\) by adding 1 to both sides: \(2x = 1\).
Divide both sides by 2 to isolate \(x\): \(x = \frac{1}{2}\).
Since the absolute value equation reduces to a single solution, verify by substituting \(x = \frac{1}{2}\) back into the original equation to ensure it satisfies the equation.
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Absolute Value Definition
The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For any expression |A| = B, if B is non-negative, then A = B or A = -B. Understanding this is crucial for solving absolute value equations.
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Vertex Form
Isolating the Absolute Value Expression
Before solving an absolute value equation, isolate the absolute value term on one side of the equation. This often involves performing inverse operations such as addition or subtraction to simplify the equation into the form |expression| = constant.
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Checking for No Solution Cases
If after isolating the absolute value, the equation takes the form |expression| = negative number, it has no solution because absolute values cannot be negative. Recognizing this helps avoid unnecessary calculations.
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5:02Solving Logarithmic Equations
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