Example $15$ . $12$ $\newline$ Find the truth set of $\newline$ $\frac{3}{4}(x+1)+1 \leq \frac{1}{2}(x-2)+5$ $\newline$ llustrate your result on the number line.

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Grade 7

Solve percent equations: word problems

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Q. Example

15

12

\newline

Find the truth set of

\newline

\frac{3}{4}(x+1)+1 \leq \frac{1}{2}(x-2)+5

\newline

llustrate your result on the number line.

Write Inequality: Write down the given inequality. $\newline$ $(\frac{3}{4})(x+1) + 1 \leq (\frac{1}{2})(x-2) + 5$
Clear Fractions: Clear the fractions by finding a common denominator, which in this case is $4$ . Multiply every term by $4$ to eliminate the fractions. $\newline$ $4 \times \left(\frac{3}{4}\right)(x+1) + 4 \times 1 \leq 4 \times \left(\frac{1}{2}\right)(x-2) + 4 \times 5$
Simplify Equation: Simplify the equation by performing the multiplication. $3(x+1) + 4 \leq 2(x-2) + 20$
Distribute Multiplication: Distribute the multiplication over addition on both sides of the inequality. $3x + 3 + 4 \leq 2x - 4 + 20$
Combine Like Terms: Combine like terms on both sides of the inequality. $3x + 7 \leq 2x + 16$
Isolate Variable Term: Isolate the variable term on one side of the inequality by subtracting $2x$ from both sides. $\newline$ $3x - 2x + 7 \leq 2x - 2x + 16$
Simplify Inequality: Simplify the inequality after subtracting. $x + 7 \leq 16$
Isolate x: Isolate x by subtracting $7$ from both sides of the inequality. $\newline$ $x + 7 - 7 \leq 16 - 7$
Illustrate on Number Line: Simplify the inequality to find the solution for $x$ . $x \leq 9$
Illustrate on Number Line: Simplify the inequality to find the solution for $x$ . $x \leq 9$ Illustrate the result on the number line. The solution set includes all numbers less than or equal to $9$ . This can be represented on the number line with a closed circle at $9$ and a line extending to the left, indicating all numbers less than or equal to $9$ are included in the truth set.