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Chapter 3: Problem 53

Solve the inequality and sketch the solution set on a number line. $$x+4<2 x-1$$

Short Answer

Expert verified
x > 5

Step by step solution

01

- Move all terms involving x to one side

Start by subtracting x from both sides of the inequality: x + 4 - x < 2x - 1 - x This simplifies to: 4 < x - 1.
02

- Isolate x

Next, add 1 to both sides of the inequality to isolate x: 4 + 1 < x - 1 + 1 This simplifies to: 5 < x.
03

- Rewrite the inequality

Rewrite the inequality in a more standard form: x > 5.
04

- Sketch the solution set on a number line

To represent the solution on a number line, draw a number line and place an open circle at 5. Shade the region to the right of 5, indicating that x can take any value greater than 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Inequalities
Inequalities in algebra help us compare quantities and express the range of possible values for variables. Solving an inequality involves finding all the values of the variable that satisfy the inequality.

Here's how we solved the given inequality: \(x+4<2x-1\).
  • First, we moved all terms involving x to one side of the inequality by performing the same operation on both sides. This helps in simplifying the inequality. We subtracted x from both sides, leading to \(4 < x - 1\).
  • Next, we isolated x by adding 1 to both sides. This gives us \(5 < x\).
  • Lastly, we rewrote the inequality in a standard form where the variable is on the left side: \(x > 5\).
This step-by-step approach ensures the inequality is simplified, making it clear what values the variable can take.
Number Line
Representing the solution of an inequality on a number line provides a visual understanding of the possible values for the variable. For the inequality \(x > 5\):
  • Draw a number line and mark the point where x equals 5.
  • Since x is greater than 5, we use an open circle at 5 to indicate that 5 is not included in the solution.
  • Shade the region to the right of 5, showing x can be any number greater than 5.
This visual representation helps to quickly identify the set of solutions. The open circle and shaded region are key elements in showing which numbers satisfy the inequality.
Isolation of Variables
Isolating the variable is a crucial step when solving inequalities. This means manipulating the equation until the variable stands alone on one side. Let's revisit our example: \(x+4<2x-1\).
  • First, we subtracted x from both sides to remove it from one side: \(x + 4 - x < 2x - 1 - x\), simplifying to \(4 < x - 1\).
  • Then, we added 1 to both sides to remove the constant term beside x: \(4 + 1 < x - 1 + 1\), which simplifies further to \(5 < x\).
  • The resulting inequality \(x > 5\) clearly shows the isolation of the variable x on one side.
These steps make it easier to identify the variable's possible values. Isolating the variable simplifies complex inequalities into more straightforward expressions, revealing the solution quickly.

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Most popular questions from this chapter

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. The length of a rectangle is 12 more than four times its width. If the perimeter of the rectangle is 134 meters, find the dimensions for the rectangle.

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. The width of a rectangle is 10 less than five times the length. If the perimeter of the rectangle is 100 yd, find the dimensions of the rectangle.

Solve the inequality and sketch the solution set on a number line. $$-1 \leq y-3 \leq 2$$

Perform the indicated operations on the given inequality. Sketch the resulting inequality on a number line. \(-2

Translate the given phrase or sentence algebraically. \(y\) is no more than 2
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