Solved on Jan 11, 2024

Solve the inequality $x-\frac{1}{3} \leq \frac{1}{2}$ and find the solution set.

STEP 1

Assumptions
1. We are solving the inequality $x-\frac{1}{3} \leq \frac{1}{2}$ .
2. We need to isolate $x$ to one side of the inequality.

STEP 2

To isolate $x$ , we will first eliminate the fraction on the left side by adding $\frac{1}{3}$ to both sides of the inequality.
$x - \frac{1}{3} + \frac{1}{3} \leq \frac{1}{2} + \frac{1}{3}$

STEP 3

Simplify both sides of the inequality by combining like terms.
$x \leq \frac{1}{2} + \frac{1}{3}$

STEP 4

To add the fractions on the right side, we need a common denominator. The least common denominator (LCD) for 2 and 3 is 6.
$x \leq \frac{3}{6} + \frac{2}{6}$

STEP 5

Now add the fractions with the common denominator.
$x \leq \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

STEP 6

The inequality is now simplified to:
$x \leq \frac{5}{6}$
This means that $x$ is less than or equal to $\frac{5}{6}$ .
The correct answer is: $x \leq \frac{5}{6}$ .

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Solve the inequality $x-\frac{1}{3} \leq \frac{1}{2}$ and find the solution set.

STEP 1

Assumptions
1. We are solving the inequality $x-\frac{1}{3} \leq \frac{1}{2}$ .
2. We need to isolate $x$ to one side of the inequality.

STEP 2

To isolate $x$ , we will first eliminate the fraction on the left side by adding $\frac{1}{3}$ to both sides of the inequality.
$x - \frac{1}{3} + \frac{1}{3} \leq \frac{1}{2} + \frac{1}{3}$

STEP 3

Simplify both sides of the inequality by combining like terms.
$x \leq \frac{1}{2} + \frac{1}{3}$

STEP 4

To add the fractions on the right side, we need a common denominator. The least common denominator (LCD) for 2 and 3 is 6.
$x \leq \frac{3}{6} + \frac{2}{6}$

STEP 5

Now add the fractions with the common denominator.
$x \leq \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

STEP 6

The inequality is now simplified to:
$x \leq \frac{5}{6}$
This means that $x$ is less than or equal to $\frac{5}{6}$ .
The correct answer is: $x \leq \frac{5}{6}$ .

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Solve the inequality x−13≤12x-\frac{1}{3} \leq \frac{1}{2}x−31​≤21​ and find the solution set.

STEP 1

Assumptions1. We are solving the inequality x−13≤12x-\frac{1}{3} \leq \frac{1}{2}x−31​≤21​.2. We need to isolate xxx to one side of the inequality.

STEP 2

To isolate xxx, we will first eliminate the fraction on the left side by adding 13\frac{1}{3}31​ to both sides of the inequality.x−13+13≤12+13x - \frac{1}{3} + \frac{1}{3} \leq \frac{1}{2} + \frac{1}{3}x−31​+31​≤21​+31​

STEP 3

Simplify both sides of the inequality by combining like terms.x≤12+13x \leq \frac{1}{2} + \frac{1}{3}x≤21​+31​

STEP 4

To add the fractions on the right side, we need a common denominator. The least common denominator (LCD) for 2 and 3 is 6.x≤36+26x \leq \frac{3}{6} + \frac{2}{6}x≤63​+62​

STEP 5

Now add the fractions with the common denominator.x≤36+26=56x \leq \frac{3}{6} + \frac{2}{6} = \frac{5}{6}x≤63​+62​=65​

STEP 6

The inequality is now simplified to:x≤56x \leq \frac{5}{6}x≤65​This means that xxx is less than or equal to 56\frac{5}{6}65​.The correct answer is: x≤56x \leq \frac{5}{6}x≤65​.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the inequality $x-\frac{1}{3} \leq \frac{1}{2}$ and find the solution set.

Assumptions
1. We are solving the inequality $x-\frac{1}{3} \leq \frac{1}{2}$ .
2. We need to isolate $x$ to one side of the inequality.

To isolate $x$ , we will first eliminate the fraction on the left side by adding $\frac{1}{3}$ to both sides of the inequality.
$x - \frac{1}{3} + \frac{1}{3} \leq \frac{1}{2} + \frac{1}{3}$

Simplify both sides of the inequality by combining like terms.
$x \leq \frac{1}{2} + \frac{1}{3}$

To add the fractions on the right side, we need a common denominator. The least common denominator (LCD) for 2 and 3 is 6.
$x \leq \frac{3}{6} + \frac{2}{6}$

Now add the fractions with the common denominator.
$x \leq \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

The inequality is now simplified to:
$x \leq \frac{5}{6}$
This means that $x$ is less than or equal to $\frac{5}{6}$ .
The correct answer is: $x \leq \frac{5}{6}$ .