Solve for $x$:
$5-x>4 x+20$

Solve the inequality 4x+20"> 5 − x > 4 x + 20 by subtracting 5 and 4 x from both sides, then divide by -5, remembering to flip the inequality sign, resulting in x < − 3 .
Analyze the given inequalities: x < 3 , -3"> x > − 3 , 3"> x > 3 , and x < − 3 .
Find the intersection of all inequalities. The inequalities x < − 3 and -3"> x > − 3 are contradictory, meaning there is no solution that satisfies both.
The solution to the inequality 4x+20"> 5 − x > 4 x + 20 is x < − 3 . The additional inequalities create a contradiction, so considering only the first inequality, the solution is x < − 3 ​ .

Explanation

Analyze the inequality We are given the inequality 4x+20"> 5 − x > 4 x + 20 . We want to solve for x .

Subtract 5 Subtract 5 from both sides of the inequality: 4x+20-5"> 5 − x − 5 > 4 x + 20 − 5 4x+15"> − x > 4 x + 15

Subtract 4x Subtract 4 x from both sides of the inequality: 4x+15-4x"> − x − 4 x > 4 x + 15 − 4 x 15"> − 5 x > 15

Divide by -5 Divide both sides by − 5 . Remember that when dividing by a negative number, we must reverse the inequality sign: − 5 − 5 x ​ < − 5 15 ​ x < − 3

Analyze given inequalities We are given the following inequalities: x < − 3 x < 3 -3"> x > − 3 3"> x > 3 x < − 3 We want to find the values of x that satisfy all these inequalities.

Find the intersection The inequality x < − 3 is the solution to the first inequality. We are also given the inequalities x < 3 , -3"> x > − 3 , 3"> x > 3 , and x < − 3 . The inequalities x < − 3 and x < 3 are redundant since x < − 3 implies x < 3 . The inequalities x < − 3 and -3"> x > − 3 are contradictory, so there is no solution that satisfies both. However, we are also given 3"> x > 3 and x < − 3 , which are contradictory. Thus, we need to find the intersection of the solution to the first inequality, x < − 3 , with the given inequalities.

Find the intersection of inequalities The inequalities are: x < − 3 x < 3 -3"> x > − 3 3"> x > 3 x < − 3 The first inequality is x < − 3 . The second inequality is x < 3 . The third inequality is -3"> x > − 3 . The fourth inequality is 3"> x > 3 . The fifth inequality is x < − 3 .
We want to find the intersection of these inequalities. The intersection of x < − 3 and x < 3 is x < − 3 . The intersection of x < − 3 and -3"> x > − 3 is the empty set. The intersection of x < − 3 and 3"> x > 3 is the empty set. The intersection of x < − 3 and x < − 3 is x < − 3 .

Final Answer Since we have x < − 3 and -3"> x > − 3 , there is no solution. However, we also have x < − 3 and 3"> x > 3 , which is also impossible. Therefore, there is no solution that satisfies all the given inequalities. However, if we only consider the first inequality 4x+20"> 5 − x > 4 x + 20 , we found that x < − 3 .


Examples
Understanding inequalities is crucial in various real-life scenarios, such as budgeting, where you need to ensure your expenses are less than your income, or in engineering, where certain parameters must remain within specific ranges to ensure safety and efficiency. For instance, if you're planning a road trip and want to keep your total fuel cost under $100, and each gallon costs $3.5, you can use inequalities to determine the maximum number of gallons you can purchase: 3.5 x < 100 , where x is the number of gallons. Solving this inequality helps you make informed decisions to stay within your budget.

Answered by GinnyAnswer | 2025-07-07