Solved on Oct 31, 2023

Find the solution set for the absolute value inequality $|5x - 6| < 12$ .

STEP 1

Assumptions1. We are given an inequality $|5x-6|<12$ . . We need to solve for $x$ .

STEP 2

The absolute value of a number is its distance from zero on the number line, and it is always positive or zero. When we have an inequality with an absolute value, we can split it into two separate inequalities one for the positive case and one for the negative case.
So, we can rewrite the given inequality as two separate inequalities $5x-6<12$ and $-(5x-6)<12$

STEP 3

Now, we will solve each inequality separately.
First, let's solve $5x-6<12$ .
To isolate $x$ , we first add6 to both sides of the inequality $5x-6+6<12+6$

STEP 4

implify the inequality $x<18$ Then, divide both sides of the inequality by to solve for $x$ $\frac{x}{}<\frac{18}{}$

STEP 5

implify to find the solution for the first inequality $x<\frac{18}{5}$ or $x<3.$

STEP 6

Now, let's solve the second inequality $-(5x-6)<12$ .
First, distribute the negative sign $-5x+6<12$ Then, subtract6 from both sides of the inequality $-5x+6-6<12-6$

STEP 7

implify the inequality $-5x<6$ Then, divide both sides of the inequality by -5. Remember, when we divide or multiply an inequality by a negative number, we must flip the inequality sign $\frac{-5x}{-5}>\frac{6}{-5}$

STEP 8

implify to find the solution for the second inequality $x>-\frac{6}{5}$ or $x>-1.2$

STEP 9

The solution to the original inequality $|5x-6|<12$ is the intersection of the solutions to the two inequalities we found. In other words, $x$ must satisfy both $x<3.6$ and $x>-.2$ .
So, the solution is $-.2<x<3.6$ .

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Find the solution set for the absolute value inequality $|5x - 6| < 12$ .

STEP 1

Assumptions1. We are given an inequality $|5x-6|<12$ . . We need to solve for $x$ .

STEP 2

STEP 3

Now, we will solve each inequality separately.
First, let's solve $5x-6<12$ .
To isolate $x$ , we first add6 to both sides of the inequality $5x-6+6<12+6$

STEP 4

implify the inequality $x<18$ Then, divide both sides of the inequality by to solve for $x$ $\frac{x}{}<\frac{18}{}$

STEP 5

implify to find the solution for the first inequality $x<\frac{18}{5}$ or $x<3.$

STEP 6

Now, let's solve the second inequality $-(5x-6)<12$ .
First, distribute the negative sign $-5x+6<12$ Then, subtract6 from both sides of the inequality $-5x+6-6<12-6$

STEP 7

implify the inequality $-5x<6$ Then, divide both sides of the inequality by -5. Remember, when we divide or multiply an inequality by a negative number, we must flip the inequality sign $\frac{-5x}{-5}>\frac{6}{-5}$

STEP 8

implify to find the solution for the second inequality $x>-\frac{6}{5}$ or $x>-1.2$

STEP 9

The solution to the original inequality $|5x-6|<12$ is the intersection of the solutions to the two inequalities we found. In other words, $x$ must satisfy both $x<3.6$ and $x>-.2$ .
So, the solution is $-.2<x<3.6$ .

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Find the solution set for the absolute value inequality ∣5x−6∣<12|5x - 6| < 12∣5x−6∣<12.

STEP 1

Assumptions1. We are given an inequality ∣5x−6∣<12|5x-6|<12∣5x−6∣<12. . We need to solve for xxx.

STEP 2

STEP 3

Now, we will solve each inequality separately.First, let's solve 5x−6<125x-6<125x−6<12.To isolate xxx, we first add6 to both sides of the inequality5x−6+6<12+65x-6+6<12+65x−6+6<12+6

STEP 4

implify the inequalityx<18x<18x<18Then, divide both sides of the inequality by to solve for xxxx<18\frac{x}{}<\frac{18}{}x​<18​

STEP 5

implify to find the solution for the first inequalityx<185x<\frac{18}{5}x<518​orx<3.x<3.x<3.

STEP 6

Now, let's solve the second inequality −(5x−6)<12-(5x-6)<12−(5x−6)<12.First, distribute the negative sign−5x+6<12-5x+6<12−5x+6<12Then, subtract6 from both sides of the inequality−5x+6−6<12−6-5x+6-6<12-6−5x+6−6<12−6

STEP 7

implify the inequality−5x<6-5x<6−5x<6Then, divide both sides of the inequality by -5. Remember, when we divide or multiply an inequality by a negative number, we must flip the inequality sign−5x−5>6−5\frac{-5x}{-5}>\frac{6}{-5}−5−5x​>−56​

STEP 8

implify to find the solution for the second inequalityx>−65x>-\frac{6}{5}x>−56​orx>−1.2x>-1.2x>−1.2

STEP 9

The solution to the original inequality ∣5x−6∣<12|5x-6|<12∣5x−6∣<12 is the intersection of the solutions to the two inequalities we found. In other words, xxx must satisfy both x<3.6x<3.6x<3.6 and x>−.2x>-.2x>−.2.So, the solution is −.2<x<3.6-.2<x<3.6−.2<x<3.6.

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Find the solution set for the absolute value inequality $|5x - 6| < 12$ .

Assumptions1. We are given an inequality $|5x-6|<12$ . . We need to solve for $x$ .

Now, we will solve each inequality separately.
First, let's solve $5x-6<12$ .
To isolate $x$ , we first add6 to both sides of the inequality $5x-6+6<12+6$

implify the inequality $x<18$ Then, divide both sides of the inequality by to solve for $x$ $\frac{x}{}<\frac{18}{}$

implify to find the solution for the first inequality $x<\frac{18}{5}$ or $x<3.$

Now, let's solve the second inequality $-(5x-6)<12$ .
First, distribute the negative sign $-5x+6<12$ Then, subtract6 from both sides of the inequality $-5x+6-6<12-6$

implify the inequality $-5x<6$ Then, divide both sides of the inequality by -5. Remember, when we divide or multiply an inequality by a negative number, we must flip the inequality sign $\frac{-5x}{-5}>\frac{6}{-5}$

implify to find the solution for the second inequality $x>-\frac{6}{5}$ or $x>-1.2$

The solution to the original inequality $|5x-6|<12$ is the intersection of the solutions to the two inequalities we found. In other words, $x$ must satisfy both $x<3.6$ and $x>-.2$ .
So, the solution is $-.2<x<3.6$ .