Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 75

Chapter 2, Problem 75

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| + 3 = 3

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

Recommended video:

5:02

Solving Logarithmic Equations

Related Practice

Textbook Question

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 7 = 2x + 7

Textbook Question

In Exercises 59–94, solve each absolute value inequality. |(2x + 2)/4| ≥ 2

Textbook Question

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

views

Textbook Question

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. $2 x^{2} - 11 x + 3 = 0$

views

Textbook Question

Solve each equation by the method of your choice. $3 x^{2} - 7 x + 1 = 0$

Textbook Question

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. $x^{2} - 4 x - 5 = 0$

views