Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 65

Capitolo 2, Problema 65

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| = 5

Guida verificata passo dopo passo

Recall that the absolute value equation $|A| = B$ can be rewritten as two separate equations: $A = B$ and $A = -B$, provided that $B \geq 0$.

Identify $A$ and $B$ in the given equation $|2x - 1| = 5$. Here, $A = 2x - 1$ and $B = 5$.

Set up the two equations based on the definition of absolute value: $2x - 1 = 5$ and $2x - 1 = -5$.

Solve each equation separately: For $2x - 1 = 5$, add 1 to both sides and then divide by 2 to isolate $x$. For $2x - 1 = -5$, add 1 to both sides and then divide by 2 to isolate $x$.

Write the two solutions you found as the solution set for the original absolute value equation.

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Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A| = B, where B ≥ 0, the equation splits into two cases: A = B or A = -B. Understanding this definition is essential to solving absolute value equations.

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Vertex Form

Solving Linear Equations

Once the absolute value equation is split into two linear equations, solving each involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. Mastery of these steps is necessary to find the values of x that satisfy the original equation.

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Solving Linear Equations with Fractions

Checking for No Solution

If the absolute value equals a negative number, the equation has no solution because absolute values cannot be negative. Additionally, after solving, it is important to verify solutions in the original equation to ensure they are valid and not extraneous.

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Restrictions on Rational Equations

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