Question 17 of 40beginner🔧 ApplyNumerical3 marks

Solve the inequality and show the graph of the solution on number line: 3x−2<2x+13x - 2 < 2x + 1

Correct Answer

We have the inequality:

3x−2<2x+13x - 2 < 2x + 1

Step 1: Transpose terms Subtracting 2x2x from both sides, we get:

3x−2x−2<13x - 2x - 2 < 1 x−2<1x - 2 < 1

Step 2: Isolate x Adding 2 to both sides, we get:

x<1+2x < 1 + 2 x<3x < 3

Thus, all real numbers less than 3 are the solutions of the given inequality. The solution set is x∈(−∞,3)x \in (-\infty, 3).

Step 3: Graphical Representation To represent the solution on a number line:

  1. Draw a number line.
  2. Mark the point 3 with an open circle (indicating 3 is not included in the solution).
  3. Draw a dark line or arrow extending to the left of 3 to represent all numbers less than 3.
Exercise: EXERCISE 5.1 | Q: 17 | (Page Page 7)
For More Understanding

Explanation

The solution follows the standard algebraic method for solving linear inequalities as demonstrated in Example 2 and Example 5 of the provided context. By collecting variable terms on one side and constant terms on the other, we isolate x. The inequality sign remains unchanged because we are dividing by a positive number. The graphical representation uses an open circle at 3 because the inequality is strictly less than (<<), similar to the interval notation (−∞,3)(-\infty, 3) used in Example 3.

Solution Steps

  1. Step 1: Subtract 2x2x from both sides: x−2<1x - 2 < 1.

  2. Step 2: Add 2 to both sides: x<3x < 3.

  3. Step 3: Identify solution set as (−∞,3)(-\infty, 3).

  4. Step 4: Draw number line with open circle at 3 and arrow pointing left.

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