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QuestionSolve the inequality 2x+54≥x+33+1\frac{2 x+5}{4} \geq \frac{x+3}{3}+1 and express the solution in interval notation.

Studdy Solution

STEP 1

Assumptions1. We are given a linear inequality x+54≥x+33+1\frac{x+5}{4} \geq \frac{x+3}{3} +1 . We need to solve for xx and express the solution in interval notation3. We also need to graph the solution set

STEP 2

First, we need to get rid of the fractions to simplify the inequality. We can do this by finding a common multiple of the denominators4 and, which is12. Multiply each term by12.
12⋅2x+54≥12⋅x++12⋅112 \cdot \frac{2x+5}{4} \geq12 \cdot \frac{x+}{} +12 \cdot1

STEP 3

implify the inequality.
6(2x+5)≥(x+3)+126(2x+5) \geq(x+3) +12

STEP 4

Expand the brackets.
12x+30≥4x+12+1212x +30 \geq4x +12 +12

STEP 5

implify the right side of the inequality.
12x+30≥4x+2412x +30 \geq4x +24

STEP 6

Subtract 4x4x from both sides of the inequality to isolate xx terms on one side.
12x−4x≥24−3012x -4x \geq24 -30

STEP 7

implify the inequality.
x≥−6x \geq -6

STEP 8

Divide both sides of the inequality by8 to solve for xx.
x≥−68x \geq \frac{-6}{8}

STEP 9

implify the fraction on the right side.
x≥−34x \geq -\frac{3}{4}This is the solution in inequality form.

STEP 10

To express the solution in interval notation, we write it as [−34,∞)[-\frac{3}{4}, \infty).

STEP 11

To graph the solution set, draw a number line and mark −34-\frac{3}{4} on it. Since the inequality is greater than or equal to, we use a closed circle at −34-\frac{3}{4} to include this value in the solution set. Draw an arrow to the right of −34-\frac{3}{4} to indicate all numbers greater than −34-\frac{3}{4}.
The solution to the inequality x+54≥x+33+\frac{x+5}{4} \geq \frac{x+3}{3} + is x≥−34x \geq -\frac{3}{4}, which is [−34,∞)[-\frac{3}{4}, \infty) in interval notation. The graph of the solution set includes −34-\frac{3}{4} and all numbers greater than −34-\frac{3}{4}.

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