Equations and systems · Equations, inequalities and systems
Inequalities
An inequality asks "for which values?" instead of "exactly how much?". This section reviews everything you already know from 3rd ESO and widens the picture: inequalities with one or two unknowns, with fractions, with absolute value. The following sections go deeper into each type.
Intervals and what it means to solve an inequality
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Write as an interval and represent on the number line the values of $x$ that satisfy the inequality $-3 \le x < 5$. You can do it on paper and send a photo with your…
Write as an interval and represent on the number line the values of $x$ that satisfy the inequality $-3 \le x < 5$.
You can do it on paper and send a photo with your answer.
A first-degree inequality is solved by rearranging terms, with one extra rule: when you multiply or divide by a negative number, the inequality sign reverses. A second-degree inequality is solved by finding the roots and studying the sign in each interval.
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Solve the following linear inequality: $4x - 5(x - 2) \ge 3x + 2$
Find the solution to the following system of inequalities: $\begin{cases} \frac{x + 1}{2} - \frac{x - 2}{3} > x - 1 \\ 3x - 4(x + 1) \ge -6 \end{cases}$
Find the solution to the following system of inequalities:
To pass a subject, a student must have a mean mark of at least $5$ marks across the three term exams. If in the first two exams they scored $3,5$ and $4,2$, what mark…
To pass a subject, a student must have a mean mark of at least $5$ marks across the three term exams. If in the first two exams they scored $3,5$ and $4,2$, what mark must they get in the third exam to pass? (The maximum mark is $10$).
As with one unknown, an inequality such as $3x-2y\leq6$ has as its solution a set of points in the plane, not a line. To check whether a point is a solution, substitute its coordinates: if the inequality holds, the point lies in the solution region.
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Check whether the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality in two variables $3x - 2y \le 4$.
Check whether the points $A(2, -1)$, $B(0, 3)$ and $C(-1, -2)$ are solutions to the inequality in two variables $3x - 2y \le 4$.
With denominators, the first step is to remove them by multiplying by the positive lowest common multiple: that way the direction of the inequality does not change.
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Solve the absolute value inequality $|x - 3| \le 5$ and express its solution using an interval.
Solve the absolute value inequality $|x - 3| \le 5$ and express its solution using an interval.