Inequalities I

When solving inequalities, keep the following facts in mind:

$a<b \iff ac<bc$ if $c>0$

$a<b \iff ac>bc$ if $c<0$

$\vert a\vert<b \iff -b<a<b$

$\vert a\vert>b \iff a<-b$ or $a>b$

Ex 1 Solve the inequality $-2<3x+1<13$.

Sol Subtracting 1 from each term gives $-3<3x<12$, and then dividing by 3 gives $-1<x<4$. Therefore the solution is given by $(-1,4)$.

Ex 2 Solve the inequality $\vert x-4\vert<5$.

Sol Writing the inequality without absolute values gives $-5<x-4<5$, and adding 4 to every term gives $-1<x<9$. Therefore $(-1,9)$ is the solution.

Pr A Solve the inequality $\frac{x}{2}-1<3x+9$.

Pr B Solve the inequality $(x+2)^2>x^2+2^2$.

Pr C Solve the double inequality $x+3<2x+8<3x+10$.

Pr D Solve the double inequality $2x-1\le 3x-5\le x+9$.

Pr 1 Solve the inequality $\vert 2x-5\vert\le11$.

Pr 2 Solve the inequality $\vert 9-2x\vert<15$.

Pr 3 Solve the inequality $x^2<9$.

Pr 4 Solve the inequality $x^2>25$.

Pr 5 Solve the inequality $3\sqrt{x}-1<5$.

Pr 6 Solve the inequality $3x+5\vert x\vert<16$.

Pr 7 Solve the inequality $\vert x^2-10\vert\le6$.

Go to Solutions.

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