Solve the inequality for real x: 37 - (3x + 5) ≥ 9x - 8(x - 3)

Question 14 of 40intermediate🔧 ApplyNumerical2 marks

Solve the inequality for real $x$ : $37 - (3x + 5) \geq 9x - 8(x - 3)$

Correct Answer

The solution set is $(-\infty, 2]$ .

Exercise: EXERCISE 5.1 | Q: 14 | (Page Page 7)

For More Understanding

Explanation

This question is from Exercise 5.1 and requires solving a linear inequality in one variable. Following the method demonstrated in Example 3 and Example 4 of the provided context, the first step is to simplify both sides of the inequality by expanding brackets. Next, we transpose terms to group the variable terms on one side and constant terms on the other. Finally, we solve for $x$ and express the solution in interval notation, ensuring the inequality sign is handled correctly.

Solution Steps

Step 1: Expand the brackets. The given inequality is $37 - (3x + 5) \geq 9x - 8(x - 3)$ . Expanding the terms, we get:
$37 - 3x - 5 \geq 9x - 8x + 24$
Simplifying further:
$32 - 3x \geq x + 24$
Step 2: Rearrange the terms. Subtract $x$ from both sides and subtract 32 from both sides to group the variables and constants:
$32 - 24 \geq x + 3x$ $8 \geq 4x$
Step 3: Solve for x. Divide both sides by 4:
$2 \geq x$
This can be written as $x \leq 2$ .
Step 4: Write the solution set. Since $x$ is a real number, all real numbers less than or equal to 2 are solutions. Thus, the solution set is $(-\infty, 2]$ .

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