Question Solve the inequality $10-2(3x-1)>6x+10$ for the variable $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are solving the inequality $10 - 2(3x - 1) > 6x + 10$ .
2. We will use the properties of inequalities to isolate the variable $x$ .
3. We will perform similar operations on both sides of the inequality to maintain the inequality's truth.

STEP 2

First, we need to distribute the $-2$ across the terms inside the parentheses on the left side of the inequality.
$10 - 2(3x - 1) > 6x + 10$

STEP 3

Apply the distributive property.
$10 - 2 \cdot 3x + 2 \cdot 1 > 6x + 10$

STEP 4

Simplify the left side of the inequality by multiplying the terms.
$10 - 6x + 2 > 6x + 10$

STEP 5

Combine like terms on the left side of the inequality.
$12 - 6x > 6x + 10$

STEP 6

To isolate the variable $x$ , we need to get all the $x$ terms on one side and the constants on the other. Let's add $6x$ to both sides of the inequality.
$12 - 6x + 6x > 6x + 10 + 6x$

STEP 7

Simplify both sides of the inequality.
$12 > 12x + 10$

STEP 8

Now, subtract $10$ from both sides of the inequality to isolate the term with $x$ on one side.
$12 - 10 > 12x + 10 - 10$

STEP 9

Simplify both sides of the inequality.
$2 > 12x$

STEP 10

To solve for $x$ , divide both sides of the inequality by $12$ .
$\frac{2}{12} > \frac{12x}{12}$

STEP 11

Simplify both sides of the inequality.
$\frac{1}{6} > x$

STEP 12

We can rewrite the inequality with $x$ on the left side.
$x < \frac{1}{6}$
The solution to the inequality is $x < \frac{1}{6}$ .

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Question Solve the inequality $10-2(3x-1)>6x+10$ for the variable $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are solving the inequality $10 - 2(3x - 1) > 6x + 10$ .
2. We will use the properties of inequalities to isolate the variable $x$ .
3. We will perform similar operations on both sides of the inequality to maintain the inequality's truth.

STEP 2

First, we need to distribute the $-2$ across the terms inside the parentheses on the left side of the inequality.
$10 - 2(3x - 1) > 6x + 10$

STEP 3

Apply the distributive property.
$10 - 2 \cdot 3x + 2 \cdot 1 > 6x + 10$

STEP 4

Simplify the left side of the inequality by multiplying the terms.
$10 - 6x + 2 > 6x + 10$

STEP 5

Combine like terms on the left side of the inequality.
$12 - 6x > 6x + 10$

STEP 6

To isolate the variable $x$ , we need to get all the $x$ terms on one side and the constants on the other. Let's add $6x$ to both sides of the inequality.
$12 - 6x + 6x > 6x + 10 + 6x$

STEP 7

Simplify both sides of the inequality.
$12 > 12x + 10$

STEP 8

Now, subtract $10$ from both sides of the inequality to isolate the term with $x$ on one side.
$12 - 10 > 12x + 10 - 10$

STEP 9

Simplify both sides of the inequality.
$2 > 12x$

STEP 10

To solve for $x$ , divide both sides of the inequality by $12$ .
$\frac{2}{12} > \frac{12x}{12}$

STEP 11

Simplify both sides of the inequality.
$\frac{1}{6} > x$

STEP 12

We can rewrite the inequality with $x$ on the left side.
$x < \frac{1}{6}$
The solution to the inequality is $x < \frac{1}{6}$ .

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Question Solve the inequality 10−2(3x−1)>6x+1010-2(3x-1)>6x+1010−2(3x−1)>6x+10 for the variable xxx.

Studdy Solution

STEP 1

Assumptions1. We are solving the inequality 10−2(3x−1)>6x+1010 - 2(3x - 1) > 6x + 1010−2(3x−1)>6x+10.2. We will use the properties of inequalities to isolate the variable xxx.3. We will perform similar operations on both sides of the inequality to maintain the inequality's truth.

STEP 2

First, we need to distribute the −2-2−2 across the terms inside the parentheses on the left side of the inequality.10−2(3x−1)>6x+1010 - 2(3x - 1) > 6x + 1010−2(3x−1)>6x+10

STEP 3

Apply the distributive property.10−2⋅3x+2⋅1>6x+1010 - 2 \cdot 3x + 2 \cdot 1 > 6x + 1010−2⋅3x+2⋅1>6x+10

STEP 4

Simplify the left side of the inequality by multiplying the terms.10−6x+2>6x+1010 - 6x + 2 > 6x + 1010−6x+2>6x+10

STEP 5

Combine like terms on the left side of the inequality.12−6x>6x+1012 - 6x > 6x + 1012−6x>6x+10

STEP 6

To isolate the variable xxx, we need to get all the xxx terms on one side and the constants on the other. Let's add 6x6x6x to both sides of the inequality.12−6x+6x>6x+10+6x12 - 6x + 6x > 6x + 10 + 6x12−6x+6x>6x+10+6x

STEP 7

Simplify both sides of the inequality.12>12x+1012 > 12x + 1012>12x+10

STEP 8

Now, subtract 101010 from both sides of the inequality to isolate the term with xxx on one side.12−10>12x+10−1012 - 10 > 12x + 10 - 1012−10>12x+10−10

STEP 9

Simplify both sides of the inequality.2>12x2 > 12x2>12x

STEP 10

To solve for xxx, divide both sides of the inequality by 121212.212>12x12\frac{2}{12} > \frac{12x}{12}122​>1212x​

STEP 11

Simplify both sides of the inequality.16>x\frac{1}{6} > x61​>x

STEP 12

We can rewrite the inequality with xxx on the left side.x<16x < \frac{1}{6}x<61​The solution to the inequality is x<16x < \frac{1}{6}x<61​.

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Question Solve the inequality $10-2(3x-1)>6x+10$ for the variable $x$ .

Assumptions
1. We are solving the inequality $10 - 2(3x - 1) > 6x + 10$ .
2. We will use the properties of inequalities to isolate the variable $x$ .
3. We will perform similar operations on both sides of the inequality to maintain the inequality's truth.

First, we need to distribute the $-2$ across the terms inside the parentheses on the left side of the inequality.
$10 - 2(3x - 1) > 6x + 10$

Apply the distributive property.
$10 - 2 \cdot 3x + 2 \cdot 1 > 6x + 10$

Simplify the left side of the inequality by multiplying the terms.
$10 - 6x + 2 > 6x + 10$

Combine like terms on the left side of the inequality.
$12 - 6x > 6x + 10$

To isolate the variable $x$ , we need to get all the $x$ terms on one side and the constants on the other. Let's add $6x$ to both sides of the inequality.
$12 - 6x + 6x > 6x + 10 + 6x$

Simplify both sides of the inequality.
$12 > 12x + 10$

Now, subtract $10$ from both sides of the inequality to isolate the term with $x$ on one side.
$12 - 10 > 12x + 10 - 10$

Simplify both sides of the inequality.
$2 > 12x$

To solve for $x$ , divide both sides of the inequality by $12$ .
$\frac{2}{12} > \frac{12x}{12}$

Simplify both sides of the inequality.
$\frac{1}{6} > x$

We can rewrite the inequality with $x$ on the left side.
$x < \frac{1}{6}$
The solution to the inequality is $x < \frac{1}{6}$ .