QuestionSolve the inequality $9|2 x-1|+4<49$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, and is always non-negative. . The inequality $9|x-1|+4<49$ is a linear absolute value inequality, and we need to find the solution set for $x$ .

STEP 2

First, we need to isolate the absolute value term on one side of the inequality. We can do this by subtracting4 from both sides of the inequality.
$9|2x-1|+4-4<49-4$

STEP 3

implify the inequality.
$9|2x-1|<45$

STEP 4

Next, we divide both sides of the inequality by9 to isolate the absolute value term.
$\frac{9|2x-1|}{9}<\frac{45}{9}$

STEP 5

implify the inequality.
$|2x-1|<5$

STEP 6

An absolute value inequality $|a|<b$ is equivalent to $-b<a<b$ . So, we can rewrite the inequality as a compound inequality.
$-5<2x-1<5$

STEP 7

We can solve the compound inequality by isolating $x$ . First, add1 to all parts of the inequality.
$-5+1<2x-1+1<5+1$

STEP 8

implify the inequality.
$-4<2x<6$

STEP 9

Finally, divide all parts of the inequality by2 to solve for $x$ .
$\frac{-4}{2}<\frac{2x}{2}<\frac{6}{2}$

STEP 10

implify the inequality to find the solution set for $x$ .
$-2<x<3$ The solution to the inequality $9|2x-|+4<49$ is $-2<x<3$ .

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