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Q69.

Expert-verified

Algebra 2

Found in: Page 115

Book edition Middle English Edition

Author(s) Carter

Pages 804 pages

ISBN 9780079039903

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Short Answer

Solve each equation. Check your solution.
$69 . |k + 7| = 3 k - 11$

The solution of the equation is $k = 9$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step-1 – Apply the concept of Absolute value

On real number line, the distance of a number from 0 is the absolute value. It is always nonnegative.

The absolute value of a function is expressed as, if x is a real number then absolute value of x is defined as.

$|x| = x$ when x is greater than or equal to 0 . In other words, the absolute value of x is x when x is either positive or zero.

$|x| = - x$ when x is less than 0 . In other words, the absolute value of x is opposite of x when x is negative.

Step-2 –Example of Absolute value

The absolute value of 5 is expressed as $|5| = 5$ and of $- 5$ is expressed as $|- 5| = 5$

Step-3 – Simplify the equation

Consider the provided equation.

$|k + 7| = 3 k - 11$

Recall the concept of absolute value and apply it.

Case 1. $k + 7 = 3 k - 11$

Subtract k from both the sides.

$\begin{array}{l} k + 7 - k = 3 k - 11 - k \\ 7 = 2 k - 11 \end{array}$

Subtract 7 from both the sides.

$\begin{array}{l} 7 - 7 = 2 k - 11 - 7 \\ 0 = 2 k - 18 \\ 2 k = 18 \\ k = 9 \end{array}$

Case 2. $k + 7 = - (3 k - 11)$

Subtract k from both the sides.

$\begin{array}{l} k + 7 - k = - 3 k + 11 - k \\ 7 = - 2 k + 11 \end{array}$

Subtract 7 from both the sides.

$\begin{array}{l} 7 - 7 = - 2 k + 11 - 7 \\ 0 = - 2 k + 4 \\ 2 k = 4 \\ k = 2 \end{array}$

Therefore, there are two possible solutions for the equation $|k + 7| = 3 k - 11$ that are $k = 9$ and $k = 2$ .

Step-4 – Verify the solutions

Substitute the value $k = 9$ in the equation $|k + 7| = 3 k - 11$ .

$\begin{array}{l} | 9 + 7 | = 3 (9) - 11 \\ | 16 | = 27 - 11 \\ 16 = 16 \end{array}$

Since, this is true so the value $k = 9$ satisfy the equation $|k + 7| = 3 k - 11$ .

Substitute the value $k = 2$ in the equation $|k + 7| = 3 k - 11$

$\begin{array}{l} | 9 + 2 | = 3 (2) - 11 \\ | 11 | = 6 - 11 \\ 11 = - 5 \end{array}$

Since, this is not true so the value $k = 2$ does not satisfy the equation $|k + 7| = 3 k - 11$ .

Thus, the solution set is $\{9\}$ .

Hence, the solution of the equation is $k = 9$ .

.

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