Question 33 of 40intermediate🔧 ApplyNumerical4 marks

Solve the inequalities and represent the solution graphically on number line: 5x+1>−24,5x−1<245x + 1 > -24, \quad 5x - 1 < 24

Correct Answer

Step 1: Solve the first inequality. 5x + 1 > -24 or 5x > -24 - 1 or 5x > -25 or x > -5

Step 2: Solve the second inequality. 5x - 1 < 24 or 5x < 24 + 1 or 5x < 25 or x < 5

Step 3: Find the common solution. From the two inequalities, we have x > -5 and x < 5. Thus, the solution is -5 < x < 5, i.e., x ∈ (-5, 5).

Step 4: Graphical representation on number line. The solution is represented by an open interval from -5 to 5. On the number line, draw an open circle at -5 and at 5, with a dark line connecting them. This indicates all real numbers greater than -5 and less than 5.

Exercise: Miscellaneous Exercise on Chapter 5 | Q: 7 | (Page Page 10)
For More Understanding

Explanation

This question involves solving a system of two linear inequalities, similar to Example 11 in the textbook context. We solve each inequality separately using standard algebraic operations (adding/subtracting terms, then dividing by positive coefficients which preserves the inequality sign). The first inequality gives x > -5, and the second gives x < 5. The common solution is the intersection of both conditions, resulting in the open interval (-5, 5). The graphical representation uses open circles at the endpoints since the inequalities are strict (< and >, not ≤\leq or ≥\geq).

Solution Steps

  1. Step 1: Solve 5x + 1 > -24 to get x > -5

  2. Step 2: Solve 5x - 1 < 24 to get x < 5

  3. Step 3: Combine both solutions to get -5 < x < 5

  4. Step 4: Represent the open interval (-5, 5) on number line with open circles at endpoints

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