Solve 23x−4≥4x+1−1. Show the graph of the solutions on number line.
Get the complete, step-by-step math solution for: "Solve (3 x-4)/(2) ≥ (x+1)/(4)-1. Show the graph of the solutions on number line.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Clear the denominators
To eliminate the fractions, we find the least common multiple (LCM) of the denominators, which are 2 and 4. The LCM is 4. We multiply every term in the inequality by 4 to clear the denominators. This simplifies the inequality significantly.
Step 2: Distribute and simplify
Next, we distribute the 2 on the left side of the inequality. On the right side, we simplify the constant terms by combining 1 and −4. This brings us closer to isolating the variable x.
Step 3: Combine like terms
We combine the constant terms on the right side of the inequality. This step further simplifies the expression, making it easier to isolate the variable x.
Step 4: Isolate the variable term
To isolate the variable x, we move all terms containing x to one side of the inequality and all constant terms to the other side. We subtract x from both sides and add 8 to both sides.
Step 5: Simplify and solve for x
After combining like terms, we have 5x≥5. To solve for x, we divide both sides of the inequality by 5. Since we are dividing by a positive number, the direction of the inequality sign remains unchanged.
Step 6: Graph the solution on a number line
The solution x≥1 means all real numbers greater than or equal to 1. On a number line, this is represented by a closed circle at 1 (indicating 1 is included) and a line extending to the right (indicating all values greater than 1).