QuestionGraph the inequality $|4x + 2| < 10$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value inequality is given by $|4x+|<10$ . . We need to find the graph of this inequality.

STEP 2

First, we need to solve the inequality for $x$ . To do this, we can split the absolute value inequality into two separate inequalities.
$4x+2<10$ and $4x+2>-10$

STEP 3

Now, we can solve each inequality for $x$ .
For the first inequality, subtract2 from both sides $x <10 -2$

STEP 4

implify the inequality $4x <8$

STEP 5

Divide both sides by4 to solve for $x$ $x <8 /4$

STEP 6

implify to get the solution for the first inequality $x <2$

STEP 7

Now, solve the second inequality. Subtract2 from both sides $4x > -10 -2$

STEP 8

implify the inequality $4x > -12$

STEP 9

Divide both sides by4 to solve for $x$ $x > -12 /4$

STEP 10

implify to get the solution for the second inequality $x > -3$

STEP 11

So, the solution to the inequality $|4x+|<10$ is $-3 < x <$ .

STEP 12

To graph this solution, we draw a number line and mark the points $-$ and $2$ . Because the inequality is strict (i.e., $<$ and $>$ , not $\leq$ or $\geq$ ), we use open circles at $-$ and $2$ .

STEP 13

The solution is all values between $-3$ and $2$ , so we shade the region of the number line between $-3$ and $2$ .
The graph of the inequality $|x+2|<10$ is a number line with an open circle at $-3$ , an open circle at $2$ , and the region between $-3$ and $2$ shaded.

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QuestionGraph the inequality $|4x + 2| < 10$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value inequality is given by $|4x+|<10$ . . We need to find the graph of this inequality.

STEP 2

First, we need to solve the inequality for $x$ . To do this, we can split the absolute value inequality into two separate inequalities.
$4x+2<10$ and $4x+2>-10$

STEP 3

Now, we can solve each inequality for $x$ .
For the first inequality, subtract2 from both sides $x <10 -2$

STEP 4

implify the inequality $4x <8$

STEP 5

Divide both sides by4 to solve for $x$ $x <8 /4$

STEP 6

implify to get the solution for the first inequality $x <2$

STEP 7

Now, solve the second inequality. Subtract2 from both sides $4x > -10 -2$

STEP 8

implify the inequality $4x > -12$

STEP 9

Divide both sides by4 to solve for $x$ $x > -12 /4$

STEP 10

implify to get the solution for the second inequality $x > -3$

STEP 11

So, the solution to the inequality $|4x+|<10$ is $-3 < x <$ .

STEP 12

To graph this solution, we draw a number line and mark the points $-$ and $2$ . Because the inequality is strict (i.e., $<$ and $>$ , not $\leq$ or $\geq$ ), we use open circles at $-$ and $2$ .

STEP 13

The solution is all values between $-3$ and $2$ , so we shade the region of the number line between $-3$ and $2$ .
The graph of the inequality $|x+2|<10$ is a number line with an open circle at $-3$ , an open circle at $2$ , and the region between $-3$ and $2$ shaded.

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QuestionGraph the inequality ∣4x+2∣<10|4x + 2| < 10∣4x+2∣<10.

Studdy Solution

STEP 1

Assumptions1. The absolute value inequality is given by ∣4x+∣<10|4x+|<10∣4x+∣<10. . We need to find the graph of this inequality.

STEP 2

First, we need to solve the inequality for xxx. To do this, we can split the absolute value inequality into two separate inequalities.4x+2<104x+2<104x+2<10and4x+2>−104x+2>-104x+2>−10

STEP 3

Now, we can solve each inequality for xxx.For the first inequality, subtract2 from both sidesx<10−2x <10 -2x<10−2

STEP 4

implify the inequality4x<84x <84x<8

STEP 5

Divide both sides by4 to solve for xxxx<8/4x <8 /4x<8/4

STEP 6

implify to get the solution for the first inequalityx<2x <2x<2

STEP 7

Now, solve the second inequality. Subtract2 from both sides4x>−10−24x > -10 -24x>−10−2

STEP 8

implify the inequality4x>−124x > -124x>−12

STEP 9

Divide both sides by4 to solve for xxxx>−12/4x > -12 /4x>−12/4

STEP 10

implify to get the solution for the second inequalityx>−3x > -3x>−3

STEP 11

So, the solution to the inequality ∣4x+∣<10|4x+|<10∣4x+∣<10 is −3<x<-3 < x <−3<x<.

STEP 12

To graph this solution, we draw a number line and mark the points −-− and 222. Because the inequality is strict (i.e., <<< and >>>, not ≤\leq≤ or ≥\geq≥), we use open circles at −-− and 222.

STEP 13

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QuestionGraph the inequality $|4x + 2| < 10$ .

Assumptions1. The absolute value inequality is given by $|4x+|<10$ . . We need to find the graph of this inequality.

First, we need to solve the inequality for $x$ . To do this, we can split the absolute value inequality into two separate inequalities.
$4x+2<10$ and $4x+2>-10$

Now, we can solve each inequality for $x$ .
For the first inequality, subtract2 from both sides $x <10 -2$

implify the inequality $4x <8$

Divide both sides by4 to solve for $x$ $x <8 /4$

implify to get the solution for the first inequality $x <2$

Now, solve the second inequality. Subtract2 from both sides $4x > -10 -2$

implify the inequality $4x > -12$

Divide both sides by4 to solve for $x$ $x > -12 /4$

implify to get the solution for the second inequality $x > -3$

So, the solution to the inequality $|4x+|<10$ is $-3 < x <$ .

To graph this solution, we draw a number line and mark the points $-$ and $2$ . Because the inequality is strict (i.e., $<$ and $>$ , not $\leq$ or $\geq$ ), we use open circles at $-$ and $2$ .