QuestionSolve the inequality: $\frac{1}{2}(x-4)-2 x \leq 5(3-x)$ .

Studdy Solution

STEP 1

Assumptions1. The problem is a linear inequality $\frac{1}{}(x-4)- x \leq5(3-x)$ . We are required to find the range of values for $x$ that satisfy this inequality.

STEP 2

First, we need to simplify the inequality by distributing and combining like terms. Let's start with the left side of the inequality. Distribute $\frac{1}{2}$ in $(x-4)$ .
$\frac{1}{2}x -2 -2x \leq5(-x)$

STEP 3

Now, combine like terms on the left side of the inequality.
$-\frac{3}{2}x -2 \leq5(3-x)$

STEP 4

Next, distribute $$ in $(3-x)$ on the right side of the inequality.
$-\frac{3}{2}x -2 \leq15 -x$

STEP 5

To make the inequality easier to solve, let's get rid of the fractions by multiplying the entire inequality by $2$ .
$-3x -4 \leq30 -10x$

STEP 6

Now, let's get all the $x$ terms on one side and the constants on the other side. We can do this by adding $10x$ to both sides and adding $4$ to both sides.
$-3x +10x \leq30 +4$

STEP 7

implify both sides of the inequality.
$7x \leq34$

STEP 8

Finally, to solve for $x$ , divide both sides of the inequality by $7$ .
$x \leq \frac{34}{7}$ So, the solution to the inequality is $x \leq \frac{34}{7}$ .

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QuestionSolve the inequality: $\frac{1}{2}(x-4)-2 x \leq 5(3-x)$ .

Studdy Solution

STEP 1

Assumptions1. The problem is a linear inequality $\frac{1}{}(x-4)- x \leq5(3-x)$ . We are required to find the range of values for $x$ that satisfy this inequality.

STEP 2

First, we need to simplify the inequality by distributing and combining like terms. Let's start with the left side of the inequality. Distribute $\frac{1}{2}$ in $(x-4)$ .
$\frac{1}{2}x -2 -2x \leq5(-x)$

STEP 3

Now, combine like terms on the left side of the inequality.
$-\frac{3}{2}x -2 \leq5(3-x)$

STEP 4

Next, distribute $$ in $(3-x)$ on the right side of the inequality.
$-\frac{3}{2}x -2 \leq15 -x$

STEP 5

To make the inequality easier to solve, let's get rid of the fractions by multiplying the entire inequality by $2$ .
$-3x -4 \leq30 -10x$

STEP 6

Now, let's get all the $x$ terms on one side and the constants on the other side. We can do this by adding $10x$ to both sides and adding $4$ to both sides.
$-3x +10x \leq30 +4$

STEP 7

implify both sides of the inequality.
$7x \leq34$

STEP 8

Finally, to solve for $x$ , divide both sides of the inequality by $7$ .
$x \leq \frac{34}{7}$ So, the solution to the inequality is $x \leq \frac{34}{7}$ .

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QuestionSolve the inequality: 12(x−4)−2x≤5(3−x)\frac{1}{2}(x-4)-2 x \leq 5(3-x)21​(x−4)−2x≤5(3−x).

Studdy Solution

STEP 1

Assumptions1. The problem is a linear inequality 1(x−4)−x≤5(3−x)\frac{1}{}(x-4)- x \leq5(3-x)1​(x−4)−x≤5(3−x). We are required to find the range of values for xxx that satisfy this inequality.

STEP 2

First, we need to simplify the inequality by distributing and combining like terms. Let's start with the left side of the inequality. Distribute 12\frac{1}{2}21​ in (x−4)(x-4)(x−4).12x−2−2x≤5(−x)\frac{1}{2}x -2 -2x \leq5(-x)21​x−2−2x≤5(−x)

STEP 3

Now, combine like terms on the left side of the inequality.−32x−2≤5(3−x)-\frac{3}{2}x -2 \leq5(3-x)−23​x−2≤5(3−x)

STEP 4

Next, distribute $$ in $(3-x)$ on the right side of the inequality.−32x−2≤15−x-\frac{3}{2}x -2 \leq15 -x−23​x−2≤15−x

STEP 5

To make the inequality easier to solve, let's get rid of the fractions by multiplying the entire inequality by 222.−3x−4≤30−10x-3x -4 \leq30 -10x−3x−4≤30−10x

STEP 6

Now, let's get all the xxx terms on one side and the constants on the other side. We can do this by adding 10x10x10x to both sides and adding 444 to both sides.−3x+10x≤30+4-3x +10x \leq30 +4−3x+10x≤30+4

STEP 7

implify both sides of the inequality.7x≤347x \leq347x≤34

STEP 8

Finally, to solve for xxx, divide both sides of the inequality by 777.x≤347x \leq \frac{34}{7}x≤734​So, the solution to the inequality is x≤347x \leq \frac{34}{7}x≤734​.

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QuestionSolve the inequality: $\frac{1}{2}(x-4)-2 x \leq 5(3-x)$ .

Assumptions1. The problem is a linear inequality $\frac{1}{}(x-4)- x \leq5(3-x)$ . We are required to find the range of values for $x$ that satisfy this inequality.

First, we need to simplify the inequality by distributing and combining like terms. Let's start with the left side of the inequality. Distribute $\frac{1}{2}$ in $(x-4)$ .
$\frac{1}{2}x -2 -2x \leq5(-x)$

Now, combine like terms on the left side of the inequality.
$-\frac{3}{2}x -2 \leq5(3-x)$

Next, distribute $$ in $(3-x)$ on the right side of the inequality.
$-\frac{3}{2}x -2 \leq15 -x$

To make the inequality easier to solve, let's get rid of the fractions by multiplying the entire inequality by $2$ .
$-3x -4 \leq30 -10x$

Now, let's get all the $x$ terms on one side and the constants on the other side. We can do this by adding $10x$ to both sides and adding $4$ to both sides.
$-3x +10x \leq30 +4$

implify both sides of the inequality.
$7x \leq34$

Finally, to solve for $x$ , divide both sides of the inequality by $7$ .
$x \leq \frac{34}{7}$ So, the solution to the inequality is $x \leq \frac{34}{7}$ .