Solve: [tex]4 x-7 \leq 13[/tex]. Express your answer in interval notation.
The solution is [tex]x \in[/tex] [ ] (Express your answer in interval notation. Use "U" to indicate a union of intervals, and "oo" to express [tex]\infty[/tex])
Answer (1)
Add 7 to both sides of the inequality: 4 x − 7 + 7 ≤ 13 + 7 , which simplifies to 4 x ≤ 20 .
Divide both sides by 4: 4 4 x ≤ 4 20 , which simplifies to x ≤ 5 .
Express the solution in interval notation: ( − ∞ , 5 ] .
The solution to the inequality 4 x − 7 ≤ 13 in interval notation is ( − ∞ , 5 ] .
Explanation
Understanding the Problem We are given the inequality 4 x − 7 ≤ 13 and we want to solve for x and express the solution in interval notation.
Isolating the x term First, we add 7 to both sides of the inequality to isolate the term with x :
4 x − 7 + 7 ≤ 13 + 7
Simplifying the Inequality This simplifies to: 4 x ≤ 20
Solving for x Next, we divide both sides of the inequality by 4 to solve for x :
4 4 x ≤ 4 20
Simplified Solution This simplifies to: x ≤ 5
Interval Notation Finally, we express the solution in interval notation. Since x can be any number less than or equal to 5, the interval notation is ( − ∞ , 5 ] .
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, when budgeting, you might have a constraint like 'spending must be less than or equal to your income.' If your income is 2000 an d yo u w an tt o a ll oc a t e i t am o n gv a r i o u se x p e n ses ( re n t , f oo d , e n t er t ainm e n t ) , e a c h e x p e n sec anb ere p rese n t e d a s a v a r iab l e , an d t h es u m o f a ll e x p e n ses m u s t s a t i s f y t h e in e q u a l i t y : re n t + f oo d + e n t er t ainm e n t \leq 2000$. Solving such inequalities helps you determine how much you can spend on each category while staying within your budget. Similarly, in physics, constraints on velocity or acceleration can be expressed and solved using inequalities.
